If I buy bitcoins from a legitimate exchange and put them in the Electrum wallet is that wallet now forever linked to me by this transaction?
No. The
address will have some form of relations to you. Bitcoins are being sent to an address, not a wallet and the addresses within a wallet is independent of each other. The address would be connected to the transaction forever but it isn't sufficient to show that whoever sent that address definitely (100%) is also the person who initiated the withdrawal from the exchange. The probability is high though.
If I send bitcoins from this Electrum wallet to a mixer then back to a second Electrum wallet owned by me is this second wallet/coins now completely anonymous?
Could be. You must ensure that the previous address would never be used again by you and all the coins (UTXO) in the address has been spent completely. While mixer could enhance your anonymity, parties can still use Blockchain analysis to try to find the mixer's address.
Electrum is generally pretty bad for privacy since whichever server you connect your client to will be able to see every address in your wallet as well as your IP address. If you want to have as much privacy as possible, you should use Bitcoin Core.