Coinomi supports a variety of tokens natively, but users can manually add virtually any ETH contract.
When tokens are transferred to your Ethereum address, you should see an entry in your Ethereum wallet for 0.00ETH received. But the specific token wallet isn't automatically added. Please follow these instructions to add it: How do I add a supported ETH token in Coinomi?
If the token received isn't natively supported, and you don't see it in the list, then you should manually add it.
While it isn't necessary to have ETH to receive tokens, every time you send tokens out will require a small amount of ETH to pay for transaction fees. Make sure you have at least 0.002 ETH to send tokens out.