Assuming 6 handed action (everyone sees the flop):

Is a flush-over-flush considered to be a "cooler" in this game? i.e., How confident should I be that I have the best hi-hand with a Jack-high flush on a non-paired board?

Set-over-set on the flop: I flop a set of kings, with an ace on the board. Is it a "cooler" if someone has two aces in the hole?

2nd-nut Low: I have a low-hand with [A 3]. Is it a "cooler" if someone has [A 2]?

No to all. And usually a Jack high flush is good most often when you do not get raised when you bet it for value. In O8 I would consider coolers to be when top set loses to middle set, when you make the top full house and lose to quads (of the pair on the board) or when you make quads and lose to straight flush after doing so. In all the situations you described, a skillful player can often (easily) discern whether or not his or her hand is good based on the player they are against and their betting patterns.

your questions are so dependent upon number of players, board texture, opponent's styles, and the sequence of bets that it is impossible to answer.

Eh, very stakes/venue dependant. At lower stakes online, or low-mid stakes live, where players are quite passive, its not uncommon to see the second and third nut flush check/called all the way down on either a paired or unpaired board.

per the OP's question -- when i hear the term "cooler" im thinking that the hand matchup is somehow egregious or at least especially uncommon -- however, both the situations you describe are pretty super-standard in 08 (especially the nut low to second nut low one)... If youre playing Jack-high flushes or the second nut low frequently, it would be pretty rare to go a whole session without one of these things happening to you several times. esp if you happen to be playing in a game that frequently takes flops 6 handed.

coolers in plo8 are more like quads over quads or str8 flush > nut flush

... or top boat v quads

Second nut low and non-nut flushes are definitely not coolers. Maybe if you both backdoor flushes you could gripe a little if you get nutted, but it still happens all the time. If you're putting in 3 bets on any street with a non nut flush, you need a good player read.

Set over set can be gross if someone plays aces passively preflop, but again, it's omaha/8, so a set isn't even all that good.

note: this is LO8 advice

Yeah I thought about the Jack high flush situation and I do see it often where I might try to bet it for value or a hand like it, and get called down instead of raised but they still show a better hand. But what I meant was when you bet the hand and get called, at least at the tables I play, they're more often paying you off out of frustration/disbelief/curiosity than timidly showing a better hand. As percentages go, I would reckon about over 60% of the time I had a jack high flush or an equivalent situation and bet it for value over the 20k hands I played so far, I took it down at least for high. I have bet a lot worse hands for value, like the 32 flush, or just top pair when the top card pairs so I have top two pair ace kicker, and got called and won with some frequency, so thats how I think what I have said holds true. Imo definately most confident about betting these kinds of hands when I have the low locked up for protection. When you have the nut low you can have a little more freedom to value bet for high I find.

Really? I think its more like this.

Hero AcAd2c6d
Action player 8c9cTdJd rainbow.

Flop Ah3cKc ... raise raise raise raise

Turn 9d ... raise raise raise raise

River Qd ....

That would be a suckout rather than a cooler

Sets of Aces over Kings is a mild cooler in an unraised pot, as the aces might have raised and there is a mild inference that you have the best set. Once you have seen that particular player limp aces, it's no longer a cooler. Set over set in general you have to bet out once with your set, and pay off players that bluff often enough to earn the call. You can fold bottom set to a reraise from any solid player - they heard you and they still think they have you beat. King high flush or worse flush is a hand that you lose with more often than not, unless the flush came runner runner on turn and river. One good recommendation was to not bet out in early position with third flush (usually queen high) or worse, and to bet out second flush but fold to most reraises.

When playing straightforward and solid players do the right thing. Fold to reraises unless you have the nuts. When playing tricky and aggressive players, you can pick a few spots to make stands, and King high flush might be one of them, but even better is to trap them and let them bet your hand for you when you do have the nuts.

Omaha is one big cooler!