PokerStars - $0.50 NL FAST (6 max) - Holdem - 6 players

Hero (BTN): 111.3 BB
SB: 101.5 BB (VPIP: 24.34, PFR: 20.11, 3Bet Preflop: 2.78, Hands: 193)
BB: 150.44 BB
UTG: 264.92 BB (VPIP: 20.93, PFR: 20.93, 3Bet Preflop: 0.00, Hands: 43)
MP: 86.12 BB (VPIP: 20.00, PFR: 20.00, 3Bet Preflop: 0.00, Hands: 5)
CO: 82.62 BB (VPIP: 0.00, PFR: 0.00, 3Bet Preflop: 0.00, Hands: 3)

SB posts SB 0.5 BB, BB posts BB 1 BB

Pre Flop: (pot: 1.5 BB) Hero has 5d 5c
fold, fold, fold, Hero raises to 2.5 BB, fold, BB raises to 6 BB, Hero calls 3.5 BB

Flop : (12.5 BB, 2 players) 7c 2c 3s
BB checks, Hero checks

Maybe I should bet but decided to check the hand, I want a cheap showdown and not interested in a big pot with just a small pair.
How would you play here and why?

Turn : (12.5 BB, 2 players) 5h
BB bets 6 BB, Hero raises to 19.14 BB, BB calls 13.14 BB

Ofc I am raising with a set here I want to get as much money in as possible hoping he has an overpair :)
Any other thoughts?

River : (50.78 BB, 2 players) 4h

One card to a straight comes in and an A gets there, don't know if villain has many bluffs or thin value bets in this spot other then perhaps a missed club draw. An A is so easy to have for him but still I think I cant fold here.
How would you play here and why?

BB bets 24.12 BB, Hero calls 24.12 BB