On troop separation, that allows you to split a single troop type into 2 stacks. So it's exactly as your screenshot shows- instead of a single stack of 5 knights, you'll have a stack of 3 and another stack of 2. It's particularly useful for units with abilities, but remember, you can only have 5 stacks maximum.
On scrolls, if I recall correctly, you still need mana to cast the scrolls. So, your mana probably was insufficient and had to regenerate. I'd have to check in-game to confirm, though. I don't actually cast from scrolls often.
On stack movement, you don't necessarily alternate turns with the computer opponent. Rather, each stack in a battle has it's own initiative value. Stacks move according to that value. So if you have 5 high initiative stacks and the comp has 5 low iniative stacks, all your stacks will have the opportunity to act before any of the comps.
The ability to wait can also affect this. A unit that would normally act at the beginning of the turn can wait to act at the end, it will then be the first to act at the beginning of the second turn, resulting in two consecutive actions (end of 1st and beginning of 2nd).
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