Do you ever avoid befriending companions for roleplay reasons?

Yep. Sometimes it's difficult but I try to RP convincingly.

I don't care for Anders but my Hawke does, so they're friends. I've had Wardens who adored/hated Wynne whereas my feelings for her are more neutral than anything else. I've even romanced characters I don't particularly like because I thought that's what my character would do. I love Vivienne but realistically I don't think my Inquisitor would get along with her.

Of course. In fact, sometimes they lead me. My red/sarcastic Hawke was planned to be rival with Fenris and (Templar) Carver – but after a rough start, he surprisingly befriended Fenris, and brought Carver to the Deep Roads. After this, they brotherly rivalry turned into a friendship. This Hawke was in rivalry with Varric (true, it was hard, because he didn't hate him, just sometimes his gossips about his private life, but secrecy he loved to be infamous.)

Almost my every Hawke in a friendly relationship with Sebastian. There's no reason to hate him, even if sometimes don't agree with him. Despite, he's not my fav – for reasons... but at the moment, for Hawke, he's just a guy.

Sera surprisingly didn't annoy my qunari mage – he befriended her easily, while my Lavellan was not able.

Yes. I would go about playing the character and if he fails to befriend someone, then so be it. I'm redoing my canon, starting with DAO. I made an intentional decision not to use gift spam to increase approval. In the end, I only wound up with friendly status with my fav girls, Leliana, Wynne, and Shale, and no one else. But even with them I never got to 100 Warm. I frequently avoid dialogue options I think my character wouldn't take for the same reason.

She's talking about Carver hanging out with mercs or smugglers, depending on which side you take, not Hawke and their merry band.

I know, but if he was a guard, he wouldn't hanging out with them because he would have a job, or at least be in a similar situation of Aveline's that's in Hawke's group.

I couldn't befriend Vivienne at all, since my Inquisitor was extremely pro-mage rebellion. To do one of her quests you need to say "I want to help the Circle" and that obviously wasn't going to happen.

I wanted to befriend Sera, but I disagreed with her too often (and she does not like that) and didn't think the leader of the Inquisition should be running around playing pranks with her.

Really, you can still get cookies if you don't do the pranks?  Because I hate doing the pranks but that doesn't mean my Inky's don't want to be her friend. 

Yes, though the first time around for any game, I usually try to get along with everyone just to see their arcs. Afterward is when I'll branch out.

In one of my more extreme examples, my Aeducan killed Zevran right away rather than trust the elf that had just tried to assassinate her. I've also had some very moral Wardens who clashed in Morrigan, especially in the beginning. She never left, but they butted heads a lot. DA2 had one of my Hawkes being abandoned by Isabela for similar reasons, and another ignored Anders as much as possible because the whole abomination thing was too much to handle. For Inquisition, Vivienne usually gets the brunt of it when my pro-freedom Inquisitors either refuse to visit or invite her but keep her at arm's length. I've also had some dismiss Sera's arrow and just never meet her, or torpedo any chance of a lasting friendship thanks to What Pride Had Wrought. Drinking from the well, respecting the temple elves and refusing to be sorry about either will do that.

That's basically how it is with my characters. My characters are generally friendly and joking with people they get along with, and civil towards people who they don't really see eye to eye with. It makes no sense for characters in high-risk positions with only so many people they can trust to alienate those who are willing to help them.

Let's take my mage Qunari Quizzie, for example. She was never part of the Circle, and as the system stood before the war sounds very oppressive to her. However, she sees the value in rebuilding and reinventing the Circles to what they could be- a place of learning and safety for mages, with changes to allow them more freedom and responsibility as they learn to use and control their magic (without the forcing them from their parents with no contact ever rule). She disagrees with Vivienne on the nature of the mage rebellion, the control of templars over mages, and the disregard for what drove mages to the war in the first place, BUT she respects Vivenne's skill and intelligence and knows that Vivienne's intentions, though she may disagree with them, are good. Therefore she is polite and respectful to Vivienne and supports her efforts to reform the Circle with others because Vivienne wants the best for mages and she largely agrees with what Vivienne wants. As a result, My QunIsitor and Vivienne are friends.

Sera is incredibly annoying period, and I just don't like her. However, again, my QunIsitor knows that she is in no position to turn down willing and useful allies, and so she is polite and friendly to Sera, which ends up with her being friends with the elf. If I don't agree with Sera (or someone else) on something, I will choose dialogue that is not aggressive, but not pandering either. This usually nets a smaller amount of approval loss that can be pretty easily made up later.

Just playing a polite and kind Inquisitor is usually enough to win friends, in my experience. 

My Dalish Inquisitor dislikes Vivienne because, in his opinion, she supports slavery, which is barely different from the elven alienages. It offers the pretense of freedom while keeping them down.