Let’s say I mint a token on the Hathor network and I want to list that on an exchange that has Hathor already listed. Would the exchange have to integrate a new wallet SDK or can they leverage existing integrations with Hathor to also get my token listed with minor code additions?

Every custom token created on Hathor is exactly the same (in terms of tech) as the native HTR token, that’s one of the reasons Hathor say the security of any custom token is the same as the native token. There is no need to code anything, which makes them all pretty much the same and with fewer risks to have vulnerabilities.  Because of that, any exchange integration can be done with minor customizations, reducing the exchange integration time and cost for any custom token built-in Hathor (if the exchange already has HTR listed)