For future inquires, include if you're using REST or SOAP.
Here are the answers to the questions that you have above.
1) How do I get the custom templates loaded into their Docusign account? Can templates be shared between totally disparate Docusign accounts? Other options I think might work are an export or import of the templates (but that is clunky).
Maybe I can use the API to upload the templates as well???
Templates can only be shared within users on the same account. There isn't a way to upload a template via the API either. You can use composite templates for a standardized template across multiple accounts (depending on SOAP/REST, the coding changes here)
2) Who’s Docusign Integrator Key do I use?
Do they need to generate one from their Docusign account and I use that one from my software? Or do I ALWAYS use my developer integrator key???
Your application will always use your Integrator Key, which will only work in Demo until you go through the Certification to get the Integrator Key promoted to production.
More info on that process here: https://www.docusign.com/developer-center/go-live/certification
3) Who’s API username and password do I use to log in via the API? I assume theirs, but I’m not a fan of needing to save a bunch of Docusign passwords in my database.
Any other options for setting this up the right way???
You will need specific permissions enabled on a user and on an account to make API calls successfully. Most integration's use a system user to login and make these calls. If you're working within multiple accounts that you do not have administration over, you will end up running into permission errors.
It sounds like you're wanting to setup a Partner integration, which does differ a little from a standard integration.