1. What operations do cost money in it?
Basically sending out transactions is the one that costs money, but it's not something you pay to Electrum, you pay it to the network/miners.
And yes, 2FA.
2. Does the Two Factor Authentication costs money?
If yes how much? and how do I pay for it?
Yes, 2FA costs money, it's a 3rd party service (Trusted Coin). The extra money is sent together with the transactions and you'll pay either for 20 tx, either for 100 tx, upfront, in the first tx from the new (let's say) batch.
This could help with more info:
https://trustedcoin.com/#/faqBatch size Per transaction price Batch price
Pay every 20 transactions: 0.000025 BTC/tx 0.0005 BTC total
Pay every 100 transactions: 0.0000125 BTC/tx 0.00125 BTC total
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I will add that imho the added security doesn't worth the cost and if you do have bigger/big funds you want to protect you may want to acquire a hardware wallet or, if you have the background/knowledge, set up a cold storage (and make sure it never goes online).