In the context that there's the Ordinals Chain merge mined with the Bitcoin blockchain, doesn't the miner earn rewards from both chains, and therefore has the choice of which coin to sell to pay for electricity bills/other expenses?
It depends on the sidechain construction. If it is like NameCoin, which created its own tokens, out of thin air, then yes, this is just abusing 21 million coins limit, and then you have separate coins. But you can also have something like RSK, where all coins have 1:1 peg with BTC, and where nothing is created out of thin air.
Does merged mining add extra work for miners
If you mean "can the hashrate increase", then the answer is yes, because those miners are just mining regular Bitcoin blocks.
Edit: If you mean, "does the miner need to choose, which chain to mine", then it is not like that. You simply mine a regular Bitcoin blocks, which are also valid in another network. More than that: you can potentially mine blocks at lower difficulty, which is what P2Pool did (they had blocks every 30 seconds with 20 times lower difficulty, so they had 20 times more blocks, but most of them were present only on P2Pool chain, and other sidechains can use the same trick if they want).
and therefore is like a block size increase?
Not really, unless you make it mandatory for a full node to follow that chain, but then, you need a successful soft-fork to do that. But in case of additional data, there are no new consensus rules behind that. Those data are just huge OP_NOPs, which means, no additional Script needs to be enforced. Which means, there is no need for full nodes to follow that data, and download it. They only stop at verifying signatures, all data behind them are posted separately, and are not processed by full nodes, because it is not needed to determine, if a transaction is valid or not.
OK, then the sidechain would be something like a cheap placerholder for Ordinals instead of storing them in a "bank vault", which is the Bitcoin blockchain. Did I understand that right?
I believe that Ordinals' main value proposition is the fact that those dick pics and fart sounds reside in the Bitcoin blockchain. They might lose their perceived value if they're "not an actual part" of Bitcoin.
If there was no incentive to mine the Ordinal Chain, then why try to find blocks there at all?
The incentive is the same as with mining Bitcoin. Which means, you can mine Bitcoin only, or mine Bitcoin, and your own chain, at the same time.
Note that every time, when you create a transaction, you can commit any data into each of your signature, without increasing the size of your transaction.
But from the miners' viewpoint, it's not the same. They're not getting any incentives from the Ordinals sidechain at all, and it would be better for them if Ordinals remain in the Bitcoin blockchain because of the higher fees. Fees which are slowly becoming the main source of miner incentive after every halving.