As a pure thought experiment....

"Fenris and Sten"

Probably they would be in agreement more or less about mages, although Sten would not be fond of all the whining about Tevinter and mages and not really taking action against them (a similar attitude he has toward elves concerning being conquered), and Fenris, being a seeming alcoholic, would not do him a favor with Sten (similar to Ogrhen). Likely would be impressed with that he knows Qunari language. 

"Isabela and Oghren"
Oghren would probably make advances and throw sexually charged remarks toward her, and Isabela would tease him, and to an extent, even she would be repelled by his lack of hygiene. She would probably see him like Mini Gamlen.

"Merrill and Shale"
Essentially, Leliana (on steroids) and Shale's party banter Shale would find her stupid and incredibly naive, and Merril would just mumble some nonsense or get offended like she does with Avelline when she points out her stupidity.

"Dorian and Alistair"

They would probably get along fairly well. Both are wiseguys, both have emotional sides (which means they probably will avoid pulling a Snarky Hawke and Viscount dead son moment) and their views aren't that far apart to cause much of an issue.
 
"Sera and Morrigan"
They will probably end up hating each other. Morrigan will see Sera as an idiot and an annoyance (pranks, similar to interactions with Dao dog). While Sera will see Morrigan as stuck up "arshole". Similar dynamic to that one with Viviene, albeit without the whole class thing going on.

"Vivienne and Zevran"
Zeveran will troll her and try to seduce her, and Viviene will make insulting and mocking remarks about him. Similar dynamic to the one he had with Morrigan.

"Wynn and Cole"
Cole is probably positively predisposed toward her, and Wynne initially is probably weary of him due to his being a spirit on the grounds that he can be a demon or at least dangerous. However, she will probably eventually warm up to him, treat him like a very young student, and try to teach him about the world and shoulds.

Fenris and Sten: respect

Isabela and Oghren: many dirty jokes and laughing

Merrill and Shale: confusion initially, an argument eventually

Dorian and Alistair: Dorian would do to Alistair what Zevran did, which is make Alistair uncomfortable on occasion.

Sera and Morrigan: Sera banter would involve threats of tearing her eyes out

Vivienne and Zevran- oof. I'd like to think the repartee would be amazing. Initially, considered Zev would pull and IB respect but reconsidered. Zevran won't crack, Viv would give up

Wynne and Cole- the hypocrite would be nicer to him than Viv was but that's about it

Really well answered so far. Honestly I thought this thread had crashed and burned so hard that it was little more than smoking rubble.....Nice to see I'm not the only one fascinated by this kind of speculation.

I'll put some more together as requested if a few more people show an interest. Solas and Anders (either pre or post Justice) is definitely an interesting one. Iron Bull and Sigrun also has some potential I think.

I think that post-Justice Anders would present an interesting dilemma for Solas. On one level he'd consider Justice to be unfairly trapped inside a mortal body for reasons which may seem questionable. He'd definitely be concerned about Justice's behavioural changes based upon is merging with the anger inside of Anders, effectively transforming him from a spirit of justice into a spirit of vengeance.

There's a very strong likelihood that Solas would try to get through to Justice, try to calm him and return him to his natural state, which ironically would probably be good for Anders too over the long term.

On the other hand Anders' actions were (at least partly) selfless. The merge occurred because Justice could no longer inhabit the body of Kristoff (for whatever reason, based on player choice) and Anders clearly believed that he might die unless a new host was provided.

It's possible that Solas might appreciate Anders' 'sacrifice' and empathise with him on that basis, however Anders' driven, single-minded, binary nature would also cause him to feel very frustrated, given Solas' more holistic world view.

Considering the relationship is not so very different to that between Flemeth/Mythal and now quite possibly Solas/Mythal (what happened at the end of DAI still needs to be clarified), I don't think Solas would necessarily disapprove. Justice wasn't trapped against his will, as could be considered the case with the corpse of Kristoff, and Anders was also a willing participant. I would be more interested in how Solas viewed their joint action in blowing up the Chantry in Kirkwall. Solas seems an "end justifies the means" sort of person and yet he does disapprove when other people sacrifice innocents for their cause (because he seems to regard everyone else bound by his concept of mortality except himself). Since, with the Asunder retcon, Anders didn't even achieve his aim of forcing revolution by his actions, I think Solas would likely feel Anders should have followed through on his actions and led the rebellion, if he still lived, and shirked his responsibilities by tamely accepting execution if he did not. In the latter case, presumably Justice simply returned to the Fade, where Solas could seek him out and recruit him to his cause. In Tevinter Nights, Fen'Harel commands a cohort of Valor and Justice spirits, so why wouldn't our Justice be one of them?  (We only have Anders' word that Justice had transformed to a spirit of vengeance in response to his own feelings of injustice caused to Circle mages and in any case, Fen'Harel was essentially reeking vengeance on the mages involved in the blood magic ritual in attempting to bind mages to their will, so it would be equally appropriate for a vengeance Justice to get involved).

Averline and Blackwall: I think she would tolerate him whilst she still thought him a Grey Warden but would be appalled when she discovered the truth. She would definitely approve of him handing himself over to face justice and likely wouldn't approve if the Inquisitor circumvents this. I think she would also regard all the southern Grey Wardens as idiots for being taken in by Erimond.  Prior to discovering the truth, I am sure she would have been happy to discuss their respective recollections of time in the military and battle tactics against various foes.

Velanna and Josephine: Difficult to say. It depends on your relationship with Velanna and if you encouraged her to drop her resentment and hostility towards humans. Josephine would likely sympathise with the loss of her sister owing to events beyond her control. Possibly Josie would arrange for some gatherings where Velanna could tell her stories and elaborate on her culture in order to educate the listeners about the Dalish.

Nathaniel and Cole: I seem to recall that Nate was a bit prickly with Justice, so he'd probably be the same with Cole initially. Then a lot would depend on whether Cole was made more human or more spirit. I have a feeling that Nate would find spirit Cole a bit creepy and not approve of him reading people's thoughts and putting ideas into their heads.

Cassandra and Sebastian: I think they would get on like a house on fire, depending on what happened with Anders. She wouldn't approve of him attacking Kirkwall for a decision taken by Hawke, not the people of that city as a whole, but I think she would sympathise with his feelings of anger in view of Elthina being like a second mother to him. In fact she could also relate to his circumstances in being the last surviving member of his immediate family and that the others died as the result of political intrigue. Sebastian would respect Cassandra as the former right hand of the Divine and heroine in her own right through saving the life of the previous Divine Beatrice from evil mages.   If Anders was executed, then Sebastian sends aid to Kirkwall and helps with its recovery, which Cassandra would approve of.   As party members, they would probably spend time talking about faith and religion to fill the time between battles.  They might also exchange notes on how Viscount Tethras is a bit of an arse.

Actually, I think this pair would create less problems than Fenris and Dorian. Sera has a fear of magic and people that wield it that Fenris would approve of. From what I recall, Sera is definitely in favour of the Circles. He isn't that invested in things "elfy", but more about how they, and others, suffer from the elite exploiting their power, particularly the elite mages in Tevinter. I don't think Fenris would be that bothered about the Jennies operating outside the law in pushing back against this or think they should be organising a rebellion (as Solas does). When left to his own devices, Fenris spends his time intercepting and killing slave traders and slave hunters, which I doubt Sera would have a problem with. So I imagine after a bit of snarky banter, in which Fenris would more than hold his own, they would end up tolerating one another, possibly even giving a bit of grudging respect. Sera might even make him an honorary Jenny and pass the word on to the other Jenny cells to assist him should he come their way.    Oh, and I definitely think they might bond at one of Varric's impromptu drinking/gambling parties.

The absolute loathing that Viv would have for Anders would be a thing to behold. 

Cassandra and Wynne could have some interesting conversations I think. 

Whilst Cassandra doesn't feature in Asunder, which is odd when you think about it since she is the right hand of the Divine, it is clear that Wynne has a line to the Divine via Leliana and so it is hard to imagine that they wouldn't have met prior to the events of that book. I imagine that they would have got along well and had some respectful dialogue/debate, particularly concerning the Rite of Tranquility. Prior to discovering the truth in DAI, I do wonder how Cassandra viewed its necessity for keeping order and whether she shared Divine Justinia's scruples about it.

You know, the more I think about it, the stranger it is that Cassandra wasn't around during the events of Asunder.    Effectively Evangeline was filling her role at the ball where Justinia was nearly assassinated and then later when the Divine met with Wynne, Rhys and Co at the Cathedral in Val Royeaux.  Where was she?

That was my initial thought but the Inquisition was only made necessary by the rebellion of the Templars, which didn't happen until the end of Asunder, and Leliana was involved with Cassandra in the hunt for Hawke, which is why neither of them was at the Conclave, so avoided being blown up. I suppose it is possible that she had already been sent to do some reconstruction work and investigations in Kirkwall. As the Circles hadn't yet fallen, then Cassandra could have been tied up sorting out the Gallows in Kirkwall. To be honest, I always thought that leaving looking into the events that lay behind the explosion for 2 years didn't make much sense. Kirkwall would have needed someone to help organise things in the chaos that followed, although that doesn't take account for a Hawke who sided with the Templars. Still, it would explain them leaving hurriedly if they mistook the arrival of Cassandra as a threat to their freedom. Even so, having dealt with the immediate aftermath, surely Cassandra would have returned to the Divine because that was her main responsibility. Even if she had delegated it to someone else, they could at least have mentioned this.

I suppose the other explanation is that when Lambert arrived to investigate the attempt of Justinia's life, he sent Cassandra away because he didn't want her undermining his work and he was also suspicious of anyone, particularly someone like Cassandra who should have been at the Divine's side and yet wasn't. In order to avoid friction, Justinia requested that Cassandra do as he instructed, whilst assuring her she still had her completed confidence. Still, it would have been helpful to have some indication of this.

I rather suspect that DG simply forgot, or chose to forget, about Cassandra in his narrative, much as he conveniently didn't make any mention of Vivienne either, despite the fact that you would have thought she would have been involved in events, bearing in mind her importance at Court. Since she wasn't with Celene, who was absent killing elves in Halamshiral on the orders of the Divine, and this was a major ball in honour of the Divine thrown by the empress, you would think Vivienne would have been there as her representative. At least PW made a passing reference to Vivienne in Masked Empire, even if she didn't feature in the actual events that took place.