Darkcheck, it sounds to me like you are very focused on protecting your hand with bets and raises which is obviously very important. But I notice a couple of possible misconceptions:
1) Check-raising bottom pair on the flop shouldn't cause villain to fold 2 broadways on an Axx flop if villain is on the button or sometimes the CO. If a villain is doing this, you can just absolutely wreck them by bluffing total air. But now you are check-raising bottom pair, top pair, and total air. You can probably get away with this for a little while. It's just important to realize that the same argument you are using for check-raising bottom pair basically means that you should be check-raising your entire range.
2) Protection is super important in LHE, but you have to compare the EV of folding out a few outs with the EV of allowing villain to continue to spew by barreling the turn and/or river. Additionally, you can sometimes profitably block bet the river after the turn checks through since any pair, no matter how weak, is often enough. I am wondering if you are thinking "Making villain fold 6 outs is +EV" instead of considering whether check-calling could be even more +EV.
Ninefingershuffle, I almost never check-raise Ax air. When I do, it's for reasons other than anything that has been discussed so far. I think you could never check-raise A hi heads up for the rest of your life and be fine. When I do it, it is against an opponent who never 3bets flops and always waits for the turn. Never 3betting flops heads up in position and always waiting for the turn is overall very strong and difficult to exploit. Tossing in an occasional check-raise with something like an ace hi flush draw or bottom pair top kicker can be a very minor exploit against these players. I especially like it against players who will freeze up with better hands and allow the turn to go check-check. I am never planning to barrel the turn when I do this. I basically just want to 1) have some Ax in my range when the turn comes an ace since an ace turn usually favors villain's range so heavily 2) freeze better hands and 3) piss my opponent off
Very strong players are able to estimate a traditional check-raising range fairly accurately. So I am either trying to confuse strong players or piss off predictable players if I ever check-raise ace hi on the flop. Overall, I prefer to use bottom pair, top kicker instead of ace hi. But I make the occasional exception if I have a wheel draw + backdoor flush draw or the nut flush draw. But even then, it is fairly rare for me.
There was a trend in 2007-2008 to check-raise and barrel with ace hi for value that I personally believe is outdated. I almost completely stopped seeing this from strong players from 2010 onwards.
The problem is that hands like ace hi and bottom pair strongly prefer to pick off spewy turn and river barrels. They also like to block bet the river after the turn checks through in hopes that we have a better hand.
Darkcheck, I usually just check-call on all of those boards. Online, I just call down like we already talked about. Live, I may make some hero folds on the river against predictable, tellbox live players. In the very rare cases that I check-raise, I am trying to piss someone off or make them think I suck. I want to be perceived as an idiot luckbox LAG rather than a savvy LAGTAG whenever possible. I have never been the type to try to randomize my play like a GTO solver, as I find it to be a bit lazy. Randomizing this stuff GTO-style is not worth much money. Paying close attention to my opponents and basing my play on their tendencies/tells is worth a lot of money.
Also, bluff raising rivers is an essential aspect of strong LHE play. If you pay close attention, you will notice that plenty of opponents fold to river raises in situations that don't seem to make much sense. They will swear up and down that they never make hero folds on these rivers, but in reality they do. I feel like it is super important to memorize those spots where you raise the river and get a fold so that you can mimick them later on with air such as ace hi on a scary rundown. Just in general, it is crucial to watch closely for spots where people make big folds. All LHE players have their sob stories about not getting paid with their monsters on the river. But these situations make me happy because I look forward to abusing these players in the future with wild bluffs. It uses to be super common for people to fold to river raises 40+% of the time online. It was a gold mine for quite awhile, but then the stupid GTO bots and solvers had to come along
But it is still pretty effective live, especially with live tells being so abundant.