Do you ever try to match characters with the weapons they might realistically choose?

It's pretty late in the overall story, but when I get the Fade-Touched Honeycomb in Trespasser, I always give it to Sera. That way, every time she attacks with her brand-new lyrium bow, she inflicts bees. (I've also used it with her armor, which is just as fun.)

I try to.

I like to give them something vaguely appropriate, so Cassandra got the Templar armour instead of Blackwall. I tried dressing Dorian in the Sentinel Plate and had to switch him to something else as he looked so ridiculous.

That said, there came a stage where we were all running around in Sha-Brytol armour, even poor Vivienne who I'm sure would never choose to wear such a thing.

Yep, I've been known to struggle along with under leveled gear for longer than necessary, for role playing reasons... and this includes my own PC. I like to gear and skill them up in ways that reflect their character/personality, as much as possible. It can be challenging and kind of like a mini game in of itself.

No, it was never a consideration for me, save for special weapons like Bianca or Howe Bow.

No, since I prefer my companions to be in optimum conditions.

Some playthroughs I will pay more attention to picking gear that would best suit their characters, but more often than not I'll take the pragmatic route.
(Unless they piss me off, then it's The Armor of Shame for you. )

But it's for reasons like this that it'd be nice to see a crafting system which divorced stats from appearance.

So much this!  When I heard about the ME:A crafting system, with augments and whatnot, I really hoped this would be the case.  With weapons, it KINDA does, since a full auto Valkyrie is EXACTLY what I've always pictured Shepard using, and I was happy as hell once that got fixed and my Ryders got to use 'em.  But alas, 'twas not the case with armor.  I mean, you could certainly do alot to customize your gear, putting a good bit of the bonuses you wanted on gear that looked good, but sometimes I always felt like the inherent stats of the armor were wasted on some builds.

As far as DA goes, to some degree yes, I try to pick weapons the characters would realistically use.  Which means, unfortunately, in DA:I no swords for the warriors.  Period.  ALL DA games have been HORRIBLE with sword scale.  Warriors wield the equivalent of swords that would historically only be wielded in two hands...as if they were arming swords.  And the hilts are incredibly off scale as well.  In DA:I, take the default daggers you start as a Rogue, lengthen the blade by a hair, and shorten the hilt to fit one hand....that's what a sword and shield warrior would carry, not something almost as tall as they are.  

At least in DA:I, one handed axes and maces tend to be better scaled, so there ya go.  May not fit them thematically, compared to promo material and whatnot, but I like to keep a little bit of realism in my fantasy, thanks.  

If it also fits my gameplay style, then yes.

When I craft armor Sera usually gets plaideweave somewhere in her armor. Except my last game. She had to have reds and oranges in her color, being a soldier of the light. And Blackwall gets Warden related gear.