3-way pot with 555 board

PokerStars, Hold'em No Limit - $0.01/$0.02 - 6 players
Hand delivered by Upswing Poker

UTG: $2.00 (100 bb)
MP: $3.53 (177 bb)
CO: $1.80 (90 bb)
BU (Hero): $2.69 (135 bb)
SB: $0.79 (40 bb)
BB: $2.00 (100 bb)

Pre-Flop: ($0.03) Hero is BTN with 7♥ 7♣
1 fold, MP raises to $0.06, 1 fold, Hero calls $0.06, SB calls $0.05, 1 fold

Flop: ($0.20) 5♠ 5♣ 5♥ (3 players)
SB checks, MP checks, Hero bets $0.06, SB calls $0.06, MP calls $0.06

Turn: ($0.38) T♠ (3 players)
SB checks, MP checks, Hero bets $0.19, SB folds, MP raises to $0.38, Hero calls $0.19

River: ($1.14) K♣ (2 players)
MP bets $1.74, BU (Hero) folds

both players check flop and I decide to bet to protect our equity. I didn't want good overcard's to see a free turn.
I bet small and both players call.

When both players call I am thinking their ranges are PP's themselves, random 5x and over-cards. On the turn both players check again...I try ro represent the Tx and get PP's like 99 and worse to fold as well as a stubborn Ax.
I am beating A-high for now but losing to 99,88 and felt I needed to protect versus overcard on the river.

On the turn villain check-raises us. we are losing to all of his value range but I was wondering if he can be doing this as spew . His stats were VPIP 38/ PFR 19/ AF 3. correct play is probrably to fold turn but I got a bit curious and was getting great odds to see a river.

River is a K and villain leads for huge overbet on the river. How often does the K hit his range ?

summary:

- I bet flop to deny equity to over-cards
- I think Villain's turn, check-raising range could be premium PP's, ATs,KTs,QTs,JTs, TT and 5x, most likely A5s,65s.
- Villain's aggression factor is 3 and he has a high VPIP, what do you think when taking these stats into account?
- On the river when villain over-bets do we think he is ever turning a Tx into a bluff since his aggression factor is high or is he likely pure value or nothing, but leading towards heavy value in this spot?

I feel I may have over-played the hand and should have perhaps checked flop and looked to see a showdown with a decent hand here)