So back around July of 2017 I moved some BTC into a paper wallet. (And a little more in August and October of 2017.) And it's been sitting there ever since. I'm now ready to move back to an exchange account, but I am wondering what else that wallet might have accumulated in the past few years. If I recall, Bitcoin Cash and Bitcoin Gold both forked from BTC in 2017 and they both 'rewarded' air drop coins,
They are not airdropped coins. They are fork of Bitcoin.
In addition to Bitcoin Cash and Bitcoin Gold, now you should have Bitcoin SV which is fork of Bitcoin Cash.
Is there one exchange that might reward any and all air drops associated with that wallet? So I can just make one transfer and earn all the rewards? Or should I have to plan on multiple transactions?
I don't think any exchange do that.
If I don't even bother trying to deal with the air drops, what happens to those coins? For example, if I transfer the paper wallet to CoinBase, they won't add the air dropped coins to my balance because they didn't support a couple of those drops. So what happens to the coins? Is someone else who sees my transfer able to go back and clean those up? Just wondering.
They won't be lost. They will remain in your wallet until you move them.
As you are talking about a paper wallet, you should have the private key of your address. That private key can be used for accessing all forked coins including Bitcoin Cash, Bitcoin SV, Bitcoin gold, etc.
For example, you can install
Electron Cash, enter your private key and access your Bitcoin Cash. Then you can easily move your coins to any address.