next questions:
1. is there any "lightweight" Bitcoin client I can use for Lightning?
Not right now, no, but there are plans to make it work
with a pruned nodeYeah, Neutrino isn't mature enough for mainnet yet.
2. How can I move my Lightning Node to another server/ip address? How can I save all channels and peers after move?
Stop LND with
lncli stop
Then backup .lnd folder
Restoring the folder should preserve your channels and peers.
You don't need to worry about changes in IP address; your node will gossip the change to your peer and other nodes in the network.