Suppose there exists Fancycoin, an ERC20 altcoin which has some functionality implemented by the EVM. It does some nontrivial things that take up gas.
According to my understanding, I can receive Fancycoin in my wallet if somebody sends it to me, without having any ether myself.
My question is: if I have Fancycoin and I want to use it to do something (maybe it checks an oracle, or does compututations with some data internal to a block, etc.) but I don't have any ether in my wallet, will I be unable to use any of the features of Fancycoin? As mentioned in the answer to the question linked within the first comment below, I probably will not be able to send Fancycoin to someone else.) However, could functions in Fancycoin requiring gas be paid in Fancycoin somehow, and not ether?
Answering the above counts as an answer to my question, but in case you want to expand: I think most generally I'm interested in how coins built on top of Ethereum interact with ether. I don't understand this.
Best Answer
The default answer would be: No, you cannot use an account without Ether for anything related to the Fancycoin, as any interaction with a standard token requires a transaction which you need to sign and broadcast to the network and conversely pay for the transaction costs in Ether (at least for the time being you can only pay fees in Ether).
The more interesting answer is: Yes, by changing the requirements somewhat. You do sign a transaction off-chain, send it to someone else (e.g. by means of a pigeon, snail mail, or smoke signals). That someone else might then be able to send that transaction and potentially get a little reward for doing so.
The above idea is not all ironed (indicated security issues remain) but the following sketch works, I just tested it in Remix with a little help of the geth web3 console: