playing 2nd pairs on flop with initiative

I personally think the best way to understand what hands should go into what ranges is by spending time to see where all the hands in your range fit in on that particular board. Also keep in mind that mixed ranges are in play and a lot of hands go in both checking and betting ranges.

In the A86 example:
Theoretically speaking I'm pretty sure mixed range are in play here for sure. K8 is a middle pair, 87 is a middle pair. More combos of K8 will be checked than bet and more combos of 87 will be bet than checked IMO. The reason to do so is by checking K8 you keep other Kx hands in opponents range alive, the weaker ones, while if you bet them they would just fold the flop. Weaker 8x hands won't improve as much while improving opponents range too in the way you want to, like K turn makes your 2 pair a whole lot better against his K pair which will now call you down while with 87 7 turn won't help him as much or really help his T9, either way you don't get value and its a lose-lose situation.

So if you had all K8 and all 87 in your opening range then on this flop i would bet like say 4 combos and check 8 while for 87 i might do the opposite, bet 8 combos and check 4.

Now if you cbet all your middle pairs and write down all your combos which bet or check you might realize that your ranges are skewed towards betting or checking.

55-22 aren't middle pairs they are weak pairs. So if my checking and betting ranges have sub divisions of "Likely future bluffs" in checking range and "bluffs" in betting range then they would fit in one of these two depending on boards and opponents ranges. When I make ranges I do these bottom pairs in the end, just as a way to balance stuff up and see where they fit in. Fitting them in both ranges can be correct and doable cause yes they can be checked to bluff on a later street if its checked through and yes they can be bluffed now to make higher equity hands fold but the thing is I've found there are better hands to fit in both those categories.

In the hands that bluff to make higher equity hands fold category I always prefer gutshots and some backdoor draws before hands like 22-55 cause the weak draws can barrel many other turns. On A86 a BDFD can barrel cleanly on 9 turns and gutshots can barrel on straights and many cards that improve them to OESD while bottom pairs can only improve by 2 outs.

Same way hands in the "likely future bluffs" category are best described as hands that are really shitty with no equity cause best to bluff a 0% equity hand on the river than on flop where you can bluff a hand with 15% equity. However 22-55 don't even figure there cause they have some equity its just tough to realize it.

So in my analysis I sort these type of hand sin the end which are sometimes bluffed and sometimes checked.

What I'm trying to say is, people might give reasons to do that might seem conflicting but they may both be right cause these hands just follow mixed ranges i.e. sometimes checked and sometimes bet.

On A96 flush draw board, my thought process goes like this:
KK, QQ are surely checked. They will x/c 1 or 2 bets.

TT and sometimes JJ are bet cause they can check, a K or Q can hit and a hand like Kx or Qx can improve and you wanna deny them free cards.

Again, some player way more advanced than me can surely argue for mixed ranges with maybe more KK and QQ being checked and few be bet while more TT be bet and few be checked and would be correct saying that too.

I don't know if I'm making any sense to you, all I can say is play around with ranges and see where what all fits. You might realize that there are better hands to fit in some categories before worrying about the really tough spots.

The reason to do so is by checking K8 you keep other Kx hands in opponents range alive, the weaker ones, while if you bet them they would just fold the flop.

that is true, but still there are Qx, Jx which can fold.
But as you say 22-55 can be bet, can be checked, then its not terrible to bet those

In the hands that bluff to make higher equity hands fold category I always prefer gutshots and some backdoor draws before hands like 22-55 cause the weak draws can barrel many other turns.

I also would prefer backdoors, but if we can bet those, I think why not, to protect equity and make fold lot of stuff and win the pot. If we do not bet, we have to practically give them up, or let opoennt catch his KQ and stuff by seeing 2 cards. Its hard to call the bets with those PPs as bluffcatchers. He even can bet his KQ then, and so now he not only has 28% equity, but also a fold equity.

So by current thinking I like betting small pairs on A86, without any info that oppoent plays back. PLus also betting those backdoors.
Btw I yesterday skipped some cbets with like KQ on A86, to keep oponents weaker Kx and Qx. WHich makes sense I guess, even when we have backdoor draw. Not finding qiuckly the exact hand, but was using this logic for sure.

TT and sometimes JJ are bet cause they can check, a K or Q can hit and a hand like Kx or Qx can improve and you wanna deny them free cards.

Do we need to worry with JJ on A96fd? Only overcards are KQ, so maybe thats too small part of range?