I'd rather 3bet pre, don't call too much from SB.
When BB squeezes I don't like flatting either, we will often be dominated by AQ/AK + we're oop. So I'd probably fold to his squeeze.
As played I'd lead turn, he probably cbets AK and TJ, and we block AQ so I'd bet/fold turn and bet/fold river small.
River idk, maybe he plays AJ like that but probably better to fold.

It's player dependant, but KQs is fine both as a call and 3bet pre (though I personally would prefer the call). It's also fine to defend vs. squeeze in this spot though it's also ok to fold vs. tight opponents given the fact that you're out of position and since your preflop calling range is narrow to begin with you don't have to defend that much to avoid being easily exploitable.

As played, I'd just bet the turn myself for value. AK is almost always betting the flop (+ we're blocking it) so we're very rarely behind here.

As played river is close-ish, but given the fact that villain should be somewhat capped after checking back on this flop texture and our hand is fairly underplayed I'd make this call.

first of all you better have no calls in the SB strategy.
on that flop i don't see a better hand than top 2 pair with who is not CBing , he bet any AQ, AK, JT

and if you call that turn , you have to call that brick river
you must have in mind vs a turn bet what you will do on river

I'd start with a 3bet preflop but calling won't be bad. I'd play it the same postflop except call the river. I don't expect to see AK+ in his range after checking back flop other than maybe the 1 combo of KK. It's true you lose to 6 combos of AQ but you're potentially ahead of some Ax like AJ that is value betting and it's hard to give an opponent a range that consists of zero bluffs as well.

I call pre, i think clearly depends on everyone strategy about calling or 3betting, i rather call and have some decent hands on my call pre on the SB if i 3bet KQs, probably im calling almost none hands.

As played, call river? now its just repeating everybodys words, but he is expected to cbet AA,AK,KK,QQ.. and the only hand that he would play like this and beat you is AQ. He can have AJ-AT. OR some another random A.

Dont check turn. Just bomb it and barrel river too.
if he raises you on turn, you can still call with tons of equity. Then re-evaluate river.