A company wants me to build them an app for free, and I have permission to use their branding while maintaining complete control over the app. What legal issues are involved, and are there boilerplate legal docs for this arrangement?

I agree with Lawrence. You need to figure out if this is a work for hire and if so, make sure that your contract with them specifies who retains the copyright ownership. Are you going to use parts of this app for other clients? If so you might want to retain copyright ownership and give them a license to use your coding for the app. You want to have, in writing, permission to use their intellectual property (text copy, trademarks, etc.) to create and market the app.

I don't know of any boilerplate docs for this type of situation; it's possible they're out there on the interwebs. You can check on sites like Legal Zoom but your best bet is to talk to an attorney and have them draw something up for you.