- no that description string is only stored locally inside your wallet file and only you can see it when opening your wallet file. in case you recovered your coins using seed and created a new wallet all the descriptions will be lost.
so no they don't affect the size of your transaction and fees you pay.

- you can include some data in your transactions using the OP_Return code which is limited to 40 bytes
https://en.bitcoin.it/wiki/OP_RETURN
https://bitcoin.stackexchange.com/questions/29554/explanation-of-what-an-op-return-transaction-looks-like
unfortunately i do not know how you can make such transactions using Electrum. you have to create them elsewhere and then only use Electrum to sign it.
here are examples: https://www.smartbit.com.au/op-returns/eternity-wall