I'm curious as to what the correct play is with AQ & AJ specifically when we are in mid-late position and we see a couple of limpers Or a recreational player who opened to $7-$10 and some people called before us. (Live 1/2)

My default play is usually to squeeze to a large amount with AQ & AJs (some of the time with AJo). Then to cbet on most flops when Heads up, else c/f when multiway (assuming we missed flop).

For example :
Pre: UTG+2 (~$200) limps, 3 callers after. Hero (~$300) on BTN with AQo raises to $16. UTG+2 calls and 1 more called.
Comments: Is the sizing correct here?

Pot= $51 (-rake).
Flop (3way): 2c 4d 9h. Check-Check. Since it's a very dry flop, hero decides to cbet $22 despite being 3way. UTG+2 folds. Next villain calls.

Pot= $95
Turn (hu) : Kc (we don't have any clubs). Villain checks. Hero bets $60.
Comments: The K turn hits my range. I can comfortably rep overpairs & AK here. However, in live 1/2, most players only think about what their 2 cards are... So should I give up OTT here?

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Was there anything I should've done differently in the example above?

16 pre is fine but a minimum, I'd go 18 cause I go 10 plus 2 per limper. The cbet I can take or leave. It won't get any better hands to fold but equity denial is worth something. Definitely barreling the turn. A good question to ask if considering a turn barrel is whose range that card helped more, here the answer is super obvious.

Regarding the example provided, nh.

You could have raised up to 20 preflop (6x +1bb for each limper) and could have bet 2/3 pot on the flop but this stuff is a little subjective and OP's line serms fine.

This is very situation specific. Some players consistently use a $7 raise for weaker hands so you can 3bet them with AQo/AJo from LP with 100bb or more. You can also fold AQo/AJo to a $10 raise from some tight players.

Also, needless to say AQo and AJo are quite different hands. I regard 3betting a small raise and 2 calls in late position with AQo as almost standard whereas I think doing that with AJo is kind of getting out of line and flatting the small raise or folding might be the better play.

These are just my views, others will have different views and every situation will be influenced by a range of different factors including our read on our opponents, game flow and stack sizes etc.