re: Poker & Checking the best hand on the river
I don't think there is an answer to this question because it depends on both your opponent and your image at the table.
Against aggressive opponents and habitual bluffers, it may be better to check. That way they can represent the flush themselves if they are bluffing, or "value" bet their pairs(which a goold player would not do in this situation).
If you are against a very good player, a small value bet may be best, since they will check most hands behind, but may still call a small bet. This has the added advantage that you could later bluff that opponent inexpensively by representing a flush with a small bet on the river. Also note, that check-calling OOP the flop and turn means either a strong draw, a monster hand, or a donk. If a good player perceives you as a good player, he/she will know they are likely beat.
Against a loose passive player(calling station), betting is definitly best. I would bet as much as I think they will call based on prior observations.
Against a maniac, checking is definitely best since they will either overplay their pairs, or bet big/all-in to bluff you out of the pot.
Let me note that a flush is very obvious in this situation. Had it been a non-obvious str8, checking may be correct even against good players since they are likely to value bet their good and marginal hands.
I would still bet against a calling station, though. When you have a money tree in your back yard, you don't let your neighbors pick the fruits.
Finally, let's say that you have 2 opponents, and let's say for the sake of simplicity that one is a calling station(CS) and the other is a maniac(M). You are OOP with respect to both players.
If the order of betting is You, M, CS, then you should check. Maniac bets, CS overcalls, now you can re-raise(perhaps min-reraise), M will at least call or move in. If M calls, CS is bound to call. If M moves-in, CS may fold, but you still got some of CS's money from the first call. If you bet instead of checking, and M moves in, CS may fold.
If the order of betting is You, CS, M, then I think it is better to bet what CS will call. M can then either overcall, or re-raise. You can now min-re-raise sandwitching CS between You and M. If you were to check, CS is likely to check behind, and when M bets, if you just call, you close the action. If you re-raise, CS may fold.