Scruffy Dude
Even if he did, there's not much chance he'd be able to maintain a 'mask' for very long before it slipped off in a moment of joy, or sadness, or temper. His emotions run too close to the surface. So no, he doesn't care much about what others think of him, so long as they don't think he's an arse.
The Merry Maiden
Merry usually wouldn't worry about projecting a particular image of herself, but the position she's in as the leader of a large rebellion demands that she present herself as cool, calm and competent. It does take a toll on her, as she's a lot warmer under that mask, and it can be exhausting keeping her uncertainties and insecurities tidily hidden away... but she's been in this situation many times before. She's lucky in that she has a fairly regal air about her, so she doesn't have to pretend, exactly. She just needs to remain professional. She also has Boxer and Glitch with her, who are akin to family; she can be her whole self around them, and lean on them for support.
Glitch
Glitch gives absolutely zero shits what anybody thinks about her. 
Boxer
Boxer actively revels in the fact that most people assume he's a big, dumb idiot with muscles for brains, and loves basking in the moment when they realise that isn't the case. He doesn't put on a front, though. He's an open book.