Long story short I have $1000 in one erc20 token contract but 0$ in ETH. Can I at least transfer these $1000 from the token to another address without ethereum asking for gas? That is: just change owner of token funds without transaction occuring on the ethereum blockchain? 10x!
No. In order to make any changes to data on the blockchain (ie, the owner of your tokens), you will need to make a transaction, and therefore pay transaction fees. Transaction fees have to be paid in Ether.