Electrum relies on servers for blockchain data but it does not trust them. They are run by random volunteers and cannot be trusted. Getting data from multiple servers and cross checking it is better than relying on any one server. The exception to this rule is if you are running your own server in which case you have a copy of the blockchain locally and don't need to trust anyone else.

You can't send Lightning payments to offline nodes. If you disappear while your Lightning payment is being routed, your channel might end up being closed uncooperatively by the other party.

It depends on the timing.
Because in 'Reverse swap', you'll just have to successfully send your lightning part of the swap to Boltz (+fee that they'll use) and then an inbound on-chain transaction will be broadcasted.
Since your part of the swap are all through lightning, it will take only a couple of seconds;
once the on-chain transaction was broadcasted, you can safely close electrum while it's waiting to be mined.

So in Reverse Swap, you only needs to wait for the on-chain transaction to be broadcasted (status: "Unconfirmed", not "Local").
At that point, it's okay to close Electrum unlike in Forward Swap where you'll have to wait for your outbound transaction to confirm before Boltz initiate to send your lightning funds.