Say you're in the BB with Ax suited facing a button raise. You call everyone else folds. You flop a FD. In general, do you want to bet (for implied odds I guess?) or check? Is the idea to get a pot building so if you hit you have a nice pot built, or do you want to check/call or check/check and see it for cheap in case you miss?

Let's say you bet the FD and villain calls and you miss the turn. Then what, bet again? Donk bet on the flop and check on the turn looks incredibly weak, right? So you do bet the turn again or just give up?

Obviously, villain's playing style matters, but if we take that out of the equation, (let's say we just sat down or don't have a good read on him) what is usually the best play?

(stack) size matters

c/r

Assuming 100bb stacks.

I we have a piece (i.e A5hh on an Kh4H5x) flop, i'm likely to bet, whereas if we just have the fd I'm more likely to check call..

Check raising in these spots is generally fairly terrible imo, as a lot of the time villain will be cbetting air and folding to our check raise with all but the very top of his range.

The answer is probably a mixed strategy of sometimes leading out, sometimes check-calling, and sometimes check-raising. I occasionally make the play of checking and calling with the intention of leading out on the turn. In the absence of a read, I probably default towards a more passive strategy.

this is a really broad question but assuming 100bb eff stacks in general (mathematically) you don't want to be x/r'ing all your fd's as this would be way too much and you would need to adapt a very difficult strategy in order to possibly make it +ev, you should be more inclined to x/r with bottom range fd's not Ax's as they still contain a bit of sdv but the board texture also matters a ton. Leading out on the flop headsup requires a good donk betting strategy which is difficult to balance.

On the turn depending on the board texture/villains bet sizing and tendencies its perfectly fine to fold vs a cbet without any equity other than making a flush since in this case you'd need to compensate your turn call with sufficient implied odds on the river which is generally hard to get in most cases especially oop.

The strategy I find most effective with fd's (in general ofc) is to call with highcard value fd's (they can still make a decent pair) and x/r non highcard value fd's, I balance this by x/r'ing almost all of my 2 pairs, and a good amount of my sets. On the turn facing a 2nd barrel vs my fd's I'll either fold or x/r especially if I have a good blocker/blockers along with my fd, I balance this by just calling the flop with some of my sets in order to x/r on them on the turn. Calling flop with fd's is fine because you'll be calling with a lot of top pair/2nd pair hands also which will balance out your river probe bluffing frequency when you miss your fd given villain checks back the turn, this is why given this strategy its fine to fold fd's with little fold equity vs a 2nd barrel.

So to answer your question we are not looking to build a pot with fd's since villain will have more equity if he continues, we are looking to build a pot with our sets/two pairs we just use fd's (bluffs) to balance our value range. Fd's play on fold equity and this is the only reason you should be aggressive with them, the reason we are more aggressive with fd's rather than weak one pair hands is because we have very little sdv but a decent amount of equity. The weaker your actual equity is the more you rely on fold equity which is why you should be SLIGHTLY more inclined to x/r an open ender rather than a flush draw on a 2tone board, I say slightly because this one is very situation dependent as reverse implied odds will come into play here but vs a villain whose turn cbet is high we should almost always opt to include open enders in our x/r'ing range on 2tone flops. Its imperative that you understand this is a default balanced strategy (blinds vs LP) strictly on 2tone flops and it may very well be -ev if used vs the wrong (unbalanced) opponents.

This is just the strategy which I adapted through trial/error (probably similar to most regs I think) its by no means the most +ev and explaining it is 100x easier than actually understanding the right spots/frequencies so in order to apply it effectively it takes practice

Do you think that it's likely that all three of the bold plays have the same ev? I don't.

Then you even throw in a fourth play of check calling the flop and leading the turn.

Here's how I look at it:

There are a few points of indifference vs any opponent. These indifference points depend on the exact profitability of your hand. The chance of your opponent correctly guessing that his hand falls on one of these points, and then mixing his play up at exactly the right frequency because of that, is quite rare. Thus there's no need to play a mixed strategy from the big blind here.

However, if I decided that my opponent was the rare case of someone attempting to play gto poker, then I'd have to mix frequencies. Even then though, I'm still looking for leaks and I only choose between two options for a few reasons:

I'm looking for leaks because everyone has them. Even experts misplay hands.

I'm only choosing between two options because of how poker action regions interact. This is the current accepted action region breakdown, which is dependent on the betting restrictions and your position:

The original post supposes out of position, so we'll go with that.

1 bet = bet or check call.

2 bets = check raise/bet call/bet fold/check call/check fold.

3 bets = bet 3 bet/check raise/bet call/bet fold/check call.

etc. I think this illustrates why you shouldn't take more than two actions in an attempt to "mix up your play." It should always be about profitability.

Wherever you see a slash (/) is the mixed strategy region. However, there are some exceptions to the above. Such as when your range is quite strong relative to your opponent's. It's much better to be the one to do the betting and I wouldn't check raise much there.

How to select which hands for which regions would require a longer post that I'm not going to get into.

Begin with whatever strategy you use for your range in this situation. Donkbetting doesn't make sense 95%+ of the time so begin with a check and depending on the board texture raise or call.

It has to be a pretty specific board texture and holecard combo for me to consider donking. I almost always check, and then I check-raise or check-call depending on what the ranges look like, how much hand equity or fold equity I think I have, and how I expect villain to proceed.

OP posited the scenario of no reads, so we have no idea of what leaks our opponent has.

My decision tree is actually much more complicated. For example, lead out on the flop is not one of the options I consider. My choices include leading out on the flop with the intention of betting the turn on any card, leading out on the flop with the intention of checking if the flush doesn't come, and leading out on the flop with the intention of checking if the flush does come. A less sophisticated player doesn't chart out his turn action until the turn card is revealed.

If I am looking for leaks, then I would ask myself if I gain more value from seeing his reaction to a bet or his reaction to a check on my part.