To be honest I didn't like this game so much at first, but it kind of gets better as you play on. IMO the whole shop grind and initial handicap aspect takes away from these sorts of games but I understand why you do it -- because some people like that sort of thing I suppose. But I'd rather go in guns-a-blazin' from the start.
I do believe all medals are obtainable, despite "Untouchable" and "The Big Combo" being such rare unlocks.
Untouchable -- well obviously don't get hit at all. Only one person's done it. Particularly aggravating as especially any of the bosses will easily hit you since you gotta get so close to hit them, and you could be far in the game, take one heart of damage, too bad start the game all over if you want the medal.
The Big Combo -- if you're keen you'll notice this isn't possible in most levels because even if you try to drag it out, you're gonna finish out the level before you even get close, and once you finish a stage your combo resets (would have been nice if it didn't reset between stages though). One possible place to do this is the first boss. Kill a bat, wait until the combo counter is almost up, kill another bat, etc... and if you've waited like this by the time you kill that 4th bat, just about the time your combo counter is almost up, another group of four arrives. But very easy to mess up the combo here, but yes here's where you'll do it if you want that medal.