E.g. only support one coin, or the other, or both?
If one can deposit on one chain and claim a private key after the mixing process that had funds in it before the network split, wouldn't it mean that that person gets thereotically a lot more than what he began with? So what's your stance on this?
For a mixer, I don't think it really matters as much as it does with an exchange or a wallet. Since they don't actually hold any of your coins except for the chips (of which you get the private keys), you could use the coins you mix pre-split in either blockchain. All you need is the private key and you get access to both of them. For those post-split, it would depend on which blockchain the market decides to use. I expect some downtime for things to settle down so they can decide on the blockchain. In the very unlikely event that the split creates two highly used chains, we would probably see functionality for both unless ChipMixer takes sides with either one, but that's not something I know the answer to.
I think it does. Chipmixer would be at a very big loss if they still continued using old private keys(generated pre-fork), and only required users to deposit coins on both chains. Either they have to ask for 2 coins for the priv key or just wiping out the balance of old private keys altogether and transferring them to a newly generated one.
Anyways it's no guarantee that such a think would even happen, i think that chipmixer has its own plans. It'll probably not affect users as much as exchanges as you said but it will affect chipmixer. So it's nothing to worry about, if anything goes wrong i'm sure Chipmixer support is always here to help.
It'd be nice to get a public statement of whats to happen IF a chain split occurs, though.