If it were me, I'd definitely get and use the hood even though it's a pain and only offers minimal protection. Any protection it offers will still be better than a UV filter.
The nice thing about wide angle lens hoods is that they are usually shallow enough that you can still cap the lens easily, even with the hood on in the usual shooting position. Might be easier with one of those "center pinch" type caps, though.
B+W filters use a pretty slim frame to begin with (just avoid stacking multiple filters)... I'd suggest trying the one you've got before shelling out the money for a slim, which for similar quality will undoubtedly be more expensive.
If you do find you need a slim filter, be aware they aren't all created equal. Some of them are "slimmed" by not having front threads, which means you can't use a standard cap on them and kinda sucks. I think all the B+W slim filters do have front threads, though.
I very rarely use "protection" filters. Yes, I have UV filters in my camera bag just in case I need them shooting in a sandstorm or something... but they see relatively little use since I started shooting digital (which filters UV already... when shooting film, a lot of it is over-sensitive to UV, so the filter got a lot more use). I often have split seconds to get a shot and no time to stop and think "Hmmmm... Is my protection filter causing flare?", remove it, stow it in my bag, then go back to shooting. 10+ years without protection other than when absolutely necessary, 50,000 to 100,000 images a year, I have yet to see a lens damaged for lack of protection. But that's probably because I virtually always use the lens hood... even on wide angle lenses.
No, a 77-62mm step down will not work on an UWA like the 10-22. It would cause very heavy vignetting. Buy a 77mm filter.