No, the fee always paid by the address account you are use for the transaction.
If using "smart contract" you can only trigger the address you are set up to create the transaction, in the end gass fee need to paid from the account who spending the transaction.
There is no way you can paid with third account(other account).
yes, this is what I need, from one address to create a transaction and pay gas, but so that the contract would perform actions for another account, as if this another address had called it.
P.S. these addresses are mine and I have private keys)