Test

you'll also see people saying you should never value bet when you're behind your opponent's calling range, which is also incorrect

OK, let me try to come up with one example of that: we're up against an opp, and in his range he has two hands which each have 26.5% equity against ours and one hand which has 100% equity against ours. He only calls with his strong hand and folds his draws. So we're crushed by his calling range. However betting can be +EV, depending on the amount we bet. Say the pot is $20. We have $5 behind. We bet. He makes a mistake when he folds a draw because he has 26.5% equity with his draw while he is facing 16.7% pot odds. So we gain from his mistake:

EV of checking down the hand:
0.6666*0.735*$20=$9.8

EV of betting:
0.6666*$20-0.3333*$5=$11.66

Is that what you had in mind?

Note that to maximize our EV, we should bet the smallest amount that will induce him to fold his draws, up to a maximum of $10.6. Beyond this, checking down the hand becomes better.