Pretend I have no reads on any hands (with reads it's pretty obvious).

Hand #1)

Party Poker
No Limit Holdem Ring game
Blinds: $0.50/$1
6 players


Stack sizes:
UTG: $116.75
UTG+1: $142.25
CO: $179.80
Button: $13.55
Hero: $114.69
BB: $145.00

Pre-flop: (6 players) Hero is SB with
UTG calls, 2 folds, Button calls, Hero raises to $6, BB folds, UTG calls, Button folds.

Flop: ($14, 2 players)
Hero checks?

Hand #2)

Party Poker
No Limit Holdem Ring game
Blinds: $0.50/$1
6 players


Stack sizes:
UTG: $99.10
UTG+1: $77.85
CO: $122.30
Button: $314.65
SB: $113.65
Hero: $147.35

Pre-flop: (6 players) Hero is BB with
3 folds, Button calls, SB folds, SB folds, Hero raises to $5, Button calls.

Flop: ($10.5, 2 players)
Hero Checks?

And lastly, Hand #3)

Party Poker
No Limit Holdem Ring game
Blinds: $0.50/$1
5 players


Stack sizes:
UTG: $357.09
CO: $219.55
Hero: $98.50
SB: $271.65
BB: $95.30

Pre-flop: (5 players) Hero is Button with
UTG calls, CO raises to $4, Hero raises to $14, 3 folds, CO calls.

Flop: ($30.5, 2 players)
CO checks, Hero checks?

In hands #1 and #2, I don't see what I push that beats me or what calls that I beat (and I'll actually get to showdown). In hand #3 I feel JJ and TT make up a large fraction of his range (as do KK and AA), and they're not going anywhere. Thoughts?

Yes - The bet is not necessairly for value (although you'd be surprised at the calls you get from TT or 22-88, or AJ even). You don't want to give a free card as an A or K will suck. You don't want to give your opponent the aggression in the hand, especially as you're OOP - he will easily be able to push you off if you're checking an underpair when one over comes frequently.

Yes - See above. You'll get called by a lot of PPs, most 8s, and other random hands.

Yes - Without reads his range here is huge. He will fold PPs that missed and are under T, he will fold SCs that missed, he will probably fold AK and AQ too. The reason you 3bet preflop a lot is because you're going to take it down a very large majority of the time with a CB on the flop. If you get called, you have 4 outs to the nuts and 6 outs to the possible best hand.

I pretty much CB any flop after I 3bet, so if I check it looks very suspicious. I'll bet with top set, or missed suited connectors, or an overpair, or a flopped straight. I do this because I 3bet with a very wide range and my J8s will miss most of the time so I rely on folding equity. If someone starts calling my 3bets and check-raising me on the flop a lot he will take the pot away, which is when you need to start adjusting to them and then it gets into playing the player, not the cards.