1) It seems all wireless mice I have seen require batteries (makes sense), although does this mean the mouse is always turned ON? Is there a switch on the side of these mice which allow them not to suck the batteries dry?
2) Is there a wireless mouse which is smart enough to turn itself off when it has not been moved for a certain amount of time?
3) The mice I have read about seem to take AAA batteries. How long do these batteries last? I would have assumed these mice would use lithium batteries instead .... wouldn't that be better?
4) Will you recommend a mouse which can do this stuff?
Most have been covered, but my 2 cents.
1) Technically, unless they have a switch, yes they are "on". (Some travel mice store the USB receiver internally, and some of these turn off when storing them, but not all.) But...
2) They don't turn off, but most go into standby and/or hibernation after some set period of time. The ones that go into these modes last longer, and those that go quickly last even longer in general, but some people have complained about not instant response. Most mice are very good at this now, though. OTOH, bad mice run out quickly (month or two) whether they are used or not.
3) AAA is almost always travel mice. AA is more often in desktop wireless mice, but also on the bigger travel mice. Some mice, OTOH, have special lithium (embedded or replaceable); these always come with a recharger. AA are cheaper, can be replaced easily, and can run for a long time; lithium last a long time, don't need to be changed (are rechargeable), but you need to wait for them to recharge vs just putting in two new normal batteries if they run out while you want to use them. Which is better?

Depends what you value.
4) I have a Logitech wireless for my laptop (rechargeable lithium), and a Microsoft (AA) at work. I like both, both last "forever". The Logitech has a switch, the Microsoft does not. The Logitech has a change level led that lights when in use, but I've never run out while even when I started using with only one level left; I only recharge when I need to about once every month or three. The Microsoft batteries last longer, about 4-6 months, but my biggest problem is I can't tell I need to change until the mouse starts acting funny.
FWIW, I used to not like wireless at first; I had too many problems and funny acting occurances. Years later (now), they've gotten a lot better, and use them at work and on my laytop (home desktop is a wired TB, though).