I count my F1 containers as 1 batch each. But I have heard of people counting their F2's as batches, since from 1 container of F1 they could make 4 different F2 flavors, each their own "batch".

I recently had to experiment with this because half of my F2's didn't fizz at all. I left them for like 2 days, opened them, and there was absolutely no pop, it was like I opened a bottle of water. I added some more sugar to them and left them for another 2 days, and 2 of them did get a lot of fizz, but the rest did not. I am not sure what I did wrong here, but I can say that it works, even though half of the ones I added sugar to didn't fizz, the other half did! So it does in fact work!

Also, at that point, I didn't care to measure how much extra sugar I put in, I just poured a little bit in each bottle.

From my understanding and experience, yes. Even if you put it in the fridge but decide that you want it to ferment longer, you can. Just keep in mind that it will take it a bit to warm up to room temperature and to ferment to the point you want it to.

I count each F1 as a batch. I usually have 2-3 F1s going at the same time, I count each of them as a batch, so that's 2-3 batches at once.

I am in the middle of my F2, some are natural flavor and some are flavored with fruit or herbs, I decided to lay them all down because I read somewhere that it helps with potential carbonation leaks, and I can tell you that none of them carbonated. They were more fizzy after the initial F1 than they are now after F2.

Though, to be fair, I left them for just 2-3 days for F2, and while it has been HOT here, my F1 is at 83 degrees constantly, maybe I should have left them longer, so I am standing them up now and going to give them a few more days.

That's my experience with laying them down, it would be interesting to know how this goes for others, maybe it just takes longer when they are laid down vs stood up?