Is there any way to send ERC20 Dai without paying gas in Ether but paying in DAI instead? I heard about some services which called relayers which presumably can help to send popular ERC20 token without you paying gas. It works like that: you sing tx for recipient and relayed address and send it to relayer. And relayer somehow is able to pay gas for you. Is it true? If so, how does it work? Is there any other ways to send Dai by paying fee in Dai?
No, every transaction on the blockchain has a gas fee. If someone managed your wallet for you, then theoretically yes, but it would be more akin to the wallet manager paying the fee for you since they manage your wallet. So no.
This articles on DAI specifies gas fees still needing to be paid. At the end of the day the miners need to be awarded
Yes, you can send such transactions - https://github.com/austintgriffith/native-meta-transactions