$1/$2 NL, live, full table. My table image at this point is likely irrelevant, perhaps on the tight side of TAG. Table overall was quite passive, mostly timid opponents, occasional deception.

Hero stack = ~$250
Villain 1 = ~$350. 60-ish y/o male. Initially played tight-passive-occasionally deceptive. He and I were engaging in friendly chatter; he was pleasant, but his play didn't seem to change gears. His preflop limping range included KQ and AJ from MP. I don't recall him having raised preflop prior to the hand in question. I believe that his preflop raising range is premium.
Villain 2 = ~$43. His image is irrelevant, but his baby stack in my opinion is relevant to the hand.

UTG folds. UTG+1 (V2) limps. Hero limps from UTG+2 with 99. V1 raises to $10 from MP. All fold to V2, who calls. Hero calls. Preflop pot = ~$30.

Flop: 9 K J. Plan: I'm confident that I'm best here, however given Villain's probable premium hand range, and given how coordinated the board was, I don't want to build a big pot with many scare cards that could fall on the turn, OOP vs pre-raiser. I intend to trap.
V2 checks. Hero checks. V1 bets $20. V2 calls. Hero calls.

Turn: T. Hehe, scare card. Plan: I prefer to keep pot as small as possible to see a river.
V2 bets all-in his remaining $13. Hero calls. V1 raises to $45. I am more likely putting V1 on AQ than I am AK/KK/JJ, because given his style, I just cannot believe he would raise anything other than a straight here. Given that range, I like my implied odds. Hero calls.

River: K

Hero? I am confident I have the best hand. Are you checking to trap river? Or betting? If so, how much?

Why are you confident you have the best hand?

I think that both checking the flop and flatting the $20 are both pretty bad. I might be able to get behind a check raise on the flop, but i'm leaning towards donking out. We want to build a big pot with a set here, and the flop is the time to do it. There are a ton of hands that both opponents could have that will call a bet on the flop, but might not bet themselves. I would not want to offer a cheap turn to two opponents with a board this wet that hits both of their ranges pretty hard.

Raise PF. Either lead flop or c/r, c/c is awful given the board texture. Your logic is backwards, you should want to build a big pot on the flop to get value from weaker made hands and draws, before a scare card comes on the turn that either kills your action or improves your opponents to a hand that beats you.

As played, lead river, if you check villain might check back Qx, but he's never folding if you lead. Pot's just under $200 and you have about $175 behind. Sizing depends on the range you assign V1. If you think he's always (or almost always) got AQ after raising the turn, then jam.

Because, as I stated, I cannot imagine V1 raising on the turn with a 4-way straight on the turn AND not holding a straight himself. The only nonstraights I'm beat by at this point are KK, JJ, and TT, but given my read on V1, I just don't think he has these. I could be wrong, of course.

Most would probably agree with you that checking the flop and flatting V1's $20 bet are "bad". But let's paint an accurate picture for everyone.
Alternate scenario - Flop: I lead out (let's say $20 myself). How does V1 respond? He calls; that's what he does. V2 comes along. Pot = ~$90. Turn: scare card. What's your plan here? In my view, if V1 bets for value, hopefully his bet is small enough that I can call with implied odds, hoping to hit full house on river. But if I lead bet again? How much? And he has probable straight now (which I had included in his range to begin with) and raises my turn lead bet....then what do you do? You see, you've created a difficult situation for yourself.

Again, I understand where you all are coming from (the rationale of building a pot on the flop). But the turn then kills my action.
I was bummed on the river. I bet $100, he thought about it for a while, looked perturbed, then folded. I was hitting myself, thinking hindsight like: "should I have bet $50? $75?". Oh well.

Checking the flop is questionable unless you're confident in a cbet. Once you get a bet and a call back to you, you simply must raise here. Half of the deck is scary on the turn.

No offense, but your idea of "keeping the pot small until the river" is simply flawed logic and will lose you money. You flop a set and miss two chances to get money in the pot. It's textbook fancy play syndrome. It's 1/2, no need to over think things.