Should I fold at any time? Over-card and paired flop looked pretty danger. Also guy played it like he had a hand. If he bet turn instead of check I would definitely fold, After he check-raised I called because of pot-ods. Then I was pot committed.

PokerStars - $0.45+$0.05|10/20 Ante 3 NL - Holdem - 8 players
Hand converted by PokerTracker 4

CO: 74.85 BB (VPIP: 21.43, PFR: 18.92, 3Bet Preflop: 9.09, Hands: 42)
BTN: 74.35 BB (VPIP: 53.85, PFR: 0.00, 3Bet Preflop: 0.00, Hands: 13)
SB: 73.85 BB (VPIP: 14.29, PFR: 15.38, 3Bet Preflop: 0.00, Hands: 14)
BB: 74.85 BB (VPIP: 0.00, PFR: -, 3Bet Preflop: -, Hands: 1)
Hero (UTG): 74.85 BB
UTG+1: 152.55 BB (VPIP: 19.47, PFR: 15.09, 3Bet Preflop: 4.76, Hands: 113)
MP: 74.85 BB (VPIP: 0.00, PFR: -, 3Bet Preflop: -, Hands: 1)
MP+1: 74.85 BB (VPIP: 0.00, PFR: -, 3Bet Preflop: -, Hands: 1)

8 players post ante of 0.15 BB, SB posts SB 0.5 BB, BB posts BB 1 BB

Pre Flop: (pot: 2.7 BB) Hero has J J

Hero raises to 4 BB, fold, fold, fold, fold, fold, fold, BB calls 3 BB

Flop: (9.7 BB, 2 players) 6 K 6
BB checks, Hero bets 5 BB, BB raises to 10 BB, Hero calls 5 BB

Turn: (29.7 BB, 2 players) 7
BB checks, Hero bets 15 BB, BB raises to 30 BB, Hero calls 15 BB

River: (89.7 BB, 2 players) 4
BB bets 30.7 BB and is all-in, Hero calls 30.7 BB and is all-in

The beginning of a tournament is about playing lots of hands and winning small pots. It's not about getting into huge pots because the EV gained by winning big pots in the beginning is not worth the risk of busting.

It's hard to say how each unknown player will play in this scenario but I don't see your style winning in the long run. You may have caught this one particular player in a bluff but I don't see this happening the vast majority of the time against other unknowns.

What I'm saying is you will lose your stack here a lot of the time because many low stakes players don't have many bluffs in their range unless you have a particular read on the player.

You caught this player here and that's really good but would another player play the same way as this opponent who you just stacked? In all cases, you have to think about what range the player's raising you with before clicking any buttons.

Yes, I can't disagree with your opinion.
So when should I fold this hand??? After he re-raised flop ???

It depends on the villain. Each player is going to play their hand differently. You need to think about the range of hands he's raising you with.

Is he raising you without a K in his hand range? What does he have if he doesn't have a K? These are some of the questions you have to ask yourself before clicking call.

Nice reply from kwanho1.

I'd like to follow by saying that I would bet-fold flop vs the unknown. After calling check raise the pot is half the size of our stack, I want to have a lock on the hand when creating a pot so big. My "could be winning" hands are in the muck when an all in pot is building.
Given that you saw the turn I would see a free river and then fold to any bet. Once you have bet-called the turn I guess you are putting him on low equity bluff and you have to call the river.
If villain is check raising flop excessively (like all the time) or tagged as "mong" then I'd play the hand as you did on this run out as I'd expect him to lead with a K or 6 on the turn and check raise again with everything else.

call the flop raise, check turn and see what happens on river imo.