Am I stuck here on River??

PS $0.01/$0.02

Crazy table, lots of raising and hands going to showdown. Never saw alot of quality hands being played. Was folding most hands due to cards as well as playing a few other tables.

Most stacks below $5.00

3 limpers in front
Hero: ( $4.75) 8,10c limps in CO.
Button folds
SB folds
BB raises to $0.06
3 calls to Hero
Hero calls.
FLOP: 9,7,4 rb
BB bets $0.10
2 calls
Hero calls.
TURN: J
BB bets $0.25
all fold
Hero raises to $1.00
BB calls
RIVER... J
BB checks...
Hero?>?>?>??????

assume villain has a 9. bet just enough for JJ99x to call.

Assume villain would bet a boat, and is a donk calling station. Shove.

What is villain's stack (if it is less than pot to shove i'd always shove here)

Most of the time villain has 9x or something like TT/QQ here and I doubt he's planning on folding so just bet 2/3-3/4 pot for value.

Villain's stack does matter here, if it's small enough then you can't bet/fold but if there's 2x pot behind then you can bet and not be committed to call a push. (obviously he's not going to c/r very often but when he does it's never a bluff!)