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2. My understand is that yes, it can, with the exception of The Temple, where no secret rooms can spawn, whether normal, Enigma, or Int. Overflow ones. (The spr_eatr rooms technically aren't secret rooms.)
3. Potentially, yes. There is nothing in the game that says secret rooms cannot generate adjacent to boss rooms, and in fact they sometimes do. Just not north of the boss room, since that space is occupied by the elevator that takes you down a floor.
= SPOILERS AHEAD =
If you have already found that cart, and you have gone down to Floor 6, and then gone down to Floor 7, called Nowhere, and defeated the final boss of that, which is Database, and then interacted with the strange entity that asks you if you wish to continue or not and make either choice (yes -> go to the next loop, no -> run ends right there), then you have permanently unlocked Loops.
Once you have unlocked Loops, the Integer Overflow cart and its associated secret room will never spawn again. Instead, you can go to the Mods screen (where you can change run settings like ship and lethality) and at the very bottom you will find a Loop setting.
If it set to off, after you beat F6 where either Machine or Monolith awaits (depending on difficulty), the run will end after the boss fight. If it set to on, any number, then you will go down to F7 where Database is. The number simply determines what loop you are on. You can select any number between 0, and the highest loop you've beaten plus one. Defeating Database will increase that maximum by one.
Basically, once you gotten the cart and beaten the stuff it unlocks once, you won't find it ever again. But you won't need to because you can just toggle whether you want the stuff it unlocks to be available in your runs, right from the hub.