The first player to act pre-flop raises to $20, and you call from the
big blind with 8♥8♠:. The flop comes Q♦ 7♣ 3♥. You check, your
opponent bets $25, and you call. The turn is the 2:clubs:. You check,
your opponent bets $50, and you call. The river is the 8♦. Do you bet
or check?

Check.

Although you just rivered the second nuts, your range is very much
condensed. By consistently checking and calling, you’ve represented a
medium-strength bluff-catcher, which is exactly what you had until
that miracle river. You should keep telling the same story by checking
again.

Your opponent, who has shown consistent aggression, has the more
polarized range and should drive the betting on the river. She should
both bluff and value bet hands that are worse than yours, after which
you can decide whether you want to call or raise.

This does not mean that she will definitely bet, only that she should
be more likely to put money into the pot if you check than if you bet.
She has more incentive to bet, with different types of hands, than she
would have to call if you bet.

I have 2 question, let's make the board Q♦7♦3♥2♣8♣ instead.
1. OTR IP retains the polar range while OOP still have a condensed one. However hands like Q9, A7 are likely to check behind while if we lead out they have to think about a call. I don't lead ever so I'm trying to figure out where I can implement it and this spot isn't clear.
Probably my reasoning is wrong, it doesn't matter if Q9 checks back, OOP still have a condensed range and IP polar, so the condesed range always benefits from bluff catching and not lead out.

1. IP had diamonds in his range but OTR are diamonds less bluffable than other hands given the fact they block OOP folding range? I don't bluff many times with missed flush draws both because I'm a nit and also because I think opponents fold less than I imagine. The problems is my thinking lacks theory behind it and it is more a "feel" play