(1) Few people transfer from other UCs to Berkeley. It will be very near impossible to transfer from UCLA to Berkeley. If you have your heart set on Berkeley (for whatever reason), it is far more manageable to attend a community college in California and try to transfer from there. CCC’s have priority for transfer, and the classes are generally easier than those at the top UC’s - so, your GPA will likely be higher, and you will have better chances of being accepted. You will be saving two years of OOS tuition by doing that as well (~100k for a UCB degree as opposed to ~200k).
(2) You should take courses that satisfy the lower-division requirements for Berkeley’s Mathematics major. So look at Berkeley’s Math major page to see what you’ll need, and then look at UCLA’s course listings to see which courses are comparable. Note - they must match. Berkeley’s calculus classes can be a bit weird (some classes have their own textbook that is specific to Berkeley - go figure), so it’s possible that not everything will transfer over. It would also be a good idea to take breadth classes, or fulfill UC reciprocity. I don’t know the specific details for fulfilling UC reciprocity - but essentially, from my understanding, completing it at one UC will satisfy all general education requirements at another UC. I don’t think it’s a requirement, but it’s good to complete.
You don’t need AP units, but you will have to take many units to compensate - taking summer sessions can offset the amount of classes you will have to take during the regular school year. So taking summer classes is a good idea - just don’t burn out.
(3) You’ll probably have to. They’re expensive, by the unit. You may be able to take CC classes over the summer (they’re much cheaper) - but you should talk to a counselor at UCLA about that.
(4) It’s for CA community college students. It satisfies all general education requirements at any UC/CSU, for most majors. If you were to attend CC, I’m not sure it would be best to complete IGETC - it may be better to complete the L&S requirements. But because you plan to attend UCLA, it won’t apply to you.
Again, if you have your heart set on Berkeley, don’t go to UCLA (I say this only because you obviously have strong reasons for wanting to attend Berkeley - I think sticking with UCLA would be ideal, for a number of reasons that I will mention only if you ask). Transferring from a CC is very possible.