What is the connection between Pathfinders and N7?

Pathfinder is probably like a Spectre in the MWG. Though with more of an exploration then a military focus. And not all Spectres were N7s (in fact only one was) so the other pathfinders are not N7s either. Though they may be from similar organizations like Asari commandos or STG.

According to the Mass Effect wiki:
"N7 is a vocational code in the Systems Alliance military. The "N" designates special forces and the "7" refers to the highest level of proficiency. It applies to Alliance personnel who have graduated from the Interplanetary Combatives Training (ICT) program."

It also seems that the N7 ranking also carries the title of "Commander". The pathfinders are supposed to be the best all-around not just for military but also for exploration, science/research and engineering. So maybe in some sense they go though N7 training but other training as well to make use of the other.

Based on what we've seen in the orientation vids, the Pathfinders are supposed to be our best hope for survival in Andromeda. They're meant to go out and blaze a trail -- investigate potentially habitable areas and ensure that they're safe for colonization. Considering that each Ark has a Pathfinder, that there's a different race on each Ark, and that N7 is an Alliance Navy ranking, I doubt that the two are synonymous. That being said, an N7 operative would surely possess the necessary skills to be a Pathfinder. Given that there are plenty of them, sending one or two off to another galaxy isn't a huge loss.

Just knowing about the training nightmares those SAS and SEAL guys go through, imagining the entry requirements of N7 makes my head and body hurt.

Even washouts of the N program get mad props.

The hell are you going on about?

N7 is a code designated to a Human Alliance military member. It arguably expanded for Shepard's use in ME2, and for bonding use for squads in ME3, but the official designation didn't change in the trilogy.

A Pathfinder doesn't have to be N7, but a N7 (Alec Ryder) was good enough to be a Pathfinder.

Now, there may be another requirement in order to be a Pathfinder. I have some ideas. But N7 are just (even since the start of ME1) considered the most generally capable folks in dangerous space environments. Getting Alec, I don't know whether this is the AI going for the best of the best, or settling for the worst of the best (considering his age), or maybe both. Maybe the whole AI has that undercurrent though. Or maybe its simply because he's the only one willing to go that would otherwise be retired and therefore allowed to go by the Alliance, and that's it.

The black armor and N7 Alec and all that is probably meant to evoke connection to the trilogy and the history and matters of the Milky Way, but the 'default' of things otherwise may be the white armor.

Once doesn't just 'drop' N7. Even Kai Leng was designed to keep a black armor (not that that necessarily means anything) and emotionally felt rivalry with another N7. When you try so hard to be at the top, you become part of it. For all we know, the identity of being N7 and the task of being Pathfinder may have conflicted somehow (directly or indirectly) for Alec in MEA.

Anyway, point is, they're not the same.

I just think that whoever is admitted into the N program will be given a much higher consideration priority for the job due to all his/her special and intense training received beforehand. You'd absolutely hate to have a Pathfinder pressing the distress button each time he/she can't figure out a way to climb over a rock or something.

I don't think N7 gives the rank of "Commander".  I think Shepard earned that rank.  Lee Riley had the rank of captain and was an N7.  David Anderson was an admiral.

N7 is special forces.  Why would it die with Shepard?  Alec Ryder is an N7 and carried it to Andromeda.  There's also some indication that one or both of his children might receive N7 armor at some point in ME:A.

Alec bites the dust somewhere while PC INHERITS the armor during the game.