How does MultiBit deal with importing private keys and rescanning? As I understand it, Bitcoin Core doesn't maintain an address index and so has to rescan the whole blockchain when you add a new private key to the wallet, which is time intensive. If MultiBit is talking to regular Core peers, how do they look up the transactions of interest given only the addresses that the MultiBit wallet controls?
MultiBit classic retrieves the whole chain (after an optional earliest use date) after installing a BIP0037 filter which allows excluding most "uninteresting" transactions. It is still pretty time intensive.
Multibit HD does not support importing Bitcoin Core wallets. It may allow importing individual keys but I'm not sure. Mostly it uses HD wallets where a single seed generates many private keys. Either way the restore process is largely the same, involving scanning of the blockchain. HD or not HD makes no difference to the actual blocks.