Would you open with the plan to 1 to pat with 2222 and 7777 from any position? Would you 3-bet those hands versus a single raiser from what positions against what positions in general? Obviously, 2222 is much the stronger hand.

From what positions would you open 227, 277, 2277, 2227, and 2777 in general? I assume the plan would usually be to d3.

Those are all steal hands imo, or BB defense unless game is so tight that they'd all fold an 8 D1. This is especially true for 7777 because even though sevens are important blockers, your opponents won't know you've seen all of them and can wake up with a very strong D2 or D1 hand of their own, leaving you calling a 3 bet out of position and drawing 4, which is terrible for you

I generally don't raise, call a raise, or 3 bet a D3 unless I'm opening in the CO or better or defending my BB v a sb open in which case, I'd definitely 3b a hand like 2277 with position, etc.

Edit: Also, when you hold a 2 AND a 7 in your hand, it is generally too strong to turn into a snow (with rare exceptions)

If you raise 7777 and get 3!, rather than d4, can't you d1 and then pat the 2nd draw, continuing with the snow? It seems like you can continue with the snow at least until your opponent pats. Even then, you can sometimes try to get him to break, knowing he never has a #4 or better. Even against 2 opponents I would snow, but be willing to fold on later draws in some situations.

I think your thought process is fine, but your not factoring in playability. If you get 3 bet and D1 v a D2, you're going to want to lead since you're supposed to be a card ahead. If villain also D1, you can go for a xr and snow, but his D1s should be smooth enough to where he'll have too much equity to fold even against a pat hand. And at least in games I play, no one is folding a smooth made 8 to a river bet

If I did play 7777 as a steal, I prefer 4bet pre, patting to barrel off and hoping for the best

Just my opinion and fair warning: There are others here much more versed in draw games