1. Yes
2. Havign never flown out of LAX I just don't know.
3. You will arrive in Terminal E: Now you have two options: You go shower in the Terminal E Business Lounge, or you take the skymetro and go through passport control to enter the shengen zone and there go take a shower in the Terminal A Business Lounge. As far as I know both offer showers, but I personally have not checked them out yet. But I don't think there are a lot of people occupying the showers, so there should be no waiting time (fingers crossed).
4. Not really sure if you can check it for free. I would just try to take it with me. (You are allowed to take two carry-ons + a personal item). The overhead bins of the CS100 are quite huge, so space should be no issue. I know that my Rimowa carry-on does fit under the seat infront of me (although you would not want to be sitting on the seat behind it in that case). What I normally do is put the big carry on in the bins and my backpack under the seat next to me (so I still have all the legroom).
5. As of now I have never had to weight my carry-on on any LX/LH flight. Only once heading to LHR ex ZRH I actually saw an employee testing a few carry ons before boarding in the waiting area. So it does happen, but never to me so far. (And I always have the small Rimowa suitcase and a rather big backpack with me, which both might be slightly over 8kg most of the time).
6. I haven't seen a LX carry-on size checker maybe ever in ZRH. Only a small hand held weight checker once about five years ago.
7. Go for 4A T-48 hours online with your LX confimration number. If you give us the date we can check the flight and by the seats still free give you tipps what seat you should go for. Moving people around does indeed only happen when there is an equipment change on longhaul (but there pretty much never is) and only on shorthaul when for example a A321 gets swapped with an A321 (yes it's the same plane, yet we got moved). So only your European segments could be influenced by that. It happened to me twice this year. But there was always an email informing me of it.