1) Is the wallet name sent to the electrum server?
2) When I add exchange rate plugin, does the amount in my balance gets sent to another website to calculate the respective value in fiat? Or only the price is fetched and then it's multiplied/divided by the balance.
In other words, does any service other than the electrum server know how much bitcoin the wallet has in the balance?
The electrum servers don't know anything about your wallet name or wallet file.
The electrum client sends a request for each address it is interested in (i.e the addresses you see in the Addresses tab).
The server replies with the transaction history for each address.
The exchange rate plugin queries the selected rate provider for the rates and does the math in the client.
If you enable "historical rates" there might be a small bit of info leakage in that you will request rates for the specific dates of historical transactions in your wallet, and the rate provider could track that list of dates. I don't see how it could correlate the dates with specific addresses though.