I don't think e.g. LTC has a supply cap though, so it's mainly just going to be DOGE.
In case of those two coins, it is just "Violate 21M Coin Limit", because you can have LTC and DOGE. You cannot have a single Scrypt-based coin, that could be turned into LTC only, or DOGE only, when you move it between those chains. In general, to fix "Violate 21M Coin Limit" problem, you need to enforce that limit. And that means, other monetary bases should simply merge into a single one (for example by gradually merging coins, and burning overprinted ones, using free-market-based-dynamically-adjusted exchange rate). That also means, you need a single monetary base, wherever it will be placed (not necessarily on BTC, but I think it is the best option so far), and then, all altcoins should be pegged into that, exactly in the same way as Lightning Network is pegged into BTC.
Also, as Paul Sztorc said:
"On Ethereum alone, there is already over 275,000 “wrapped BTC”. Bitcoin will become multichain, on Altcoins if not on Bitcoin". Currently, we have federations, instead of decentralized sidechains (because federations are always possible). And we have some proposals, like BIP-300 and BIP-301, that were initially rejected. But, as long as altcoins exist, they will try to become multichain, and test sidechains in practice. And guess what: if any such attempt will be successful, then some altcoin could be "the base for peg-ins and peg-outs", instead of BTC. So far, many attempts are too centralized, as it is the case for "wrapped BTC", but I wonder how long it will take for people to truly understand, what decentralized chains are about, and how to implement them.
I wonder what will happen if/when the altcoins used for merge mining become less profitable for that purpose for whatever reason, despite infinite supply caps or whatever
This problem exists only because the peg is not as strong, as it is for example in Lightning Network. If you have 1:1 peg, then mining some sidechain is never "less profitable", because you mine everything, based on exactly the same difficulty. It is also when you can see, why NameCoin implemented Merged Mining wrong, what are their mistakes, and how other coins repeated them, like a parrot. Imagine NameCoin, that produces no new coins at all, where people could just sign their Bitcoins, and use them directly to push any names into NameCoin. Or imagine Litecoin, that uses SHA-256 for mining, and acts like a P2Pool, where all coins are pegged, and when you can mine some smaller amounts, do a lot of transactions there, and later batch the whole traffic, and finalize on the main chain.
Even more: if you have Lightning Network, you can try to introduce mining there. Then, instead of using millisatoshis, to pay for routing your transaction, you can pay with your computing power. And then, you don't need other mining algorithms than SHA-256 (because they are needed only if you have a separate monetary base, then you need to protect it with different algorithm). But even if you want to experiment with Scrypt, or other algos, then you can still have nodes accepting that, and use Scrypt hashes to pay for your transaction, instead of using millisatoshis.
Who is account of transaction fees?
Initially, miners could receive transaction fees directly. Later, centralized mining pools were created, and currently they are the ones receiving the coinbase reward. However, inside other networks, like Lightning Network, routing nodes receive all fees, and mainchain miners can get some, only when you open or close some channel. In sidechains, miners can always receive fees in P2P way, because they can mine the mainchain directly, or use some decentralized sidechain, mine some coins there (with Merged Mining, so it will raise mainchain difficulty as well), batch all of that, and withdraw it on a mainchain, when they mined something that is worth moving there.
In general, I think the question about fees is quite important, because a lot of topics were born from handling that incorrectly (for example, almost all altcoins picked "Violate 21M Coin Limit" as their solution).