Nope -- they agree to nothing more than the ability to use your service and that may require access to their information (and that MySpace is not liable for anything).
It's also important to note that what you propose above can't be done, either. You said "search our DB" -- I assume you mean a database you maintain offsite. You can't store DisplayNames off-site. You can cache them, for that user's convienence, for up to 24 hours, but you can't store them. That's not a MySpace limitation - it's the law; those users legally own those names; keeping a duplicate is tantamount to copyright theft.
The only I can think of, and I'm not sure how MySpace will handle this (and it applies to the above solution, as well, I guess) is to have the user agree to a Terms of Service or Usage for your application before they start using your service. I honestly don't know if MySpace will allow you to extend your own Terms to it's users. I haven't seen anything that says you can't (as long as your Terms do not conflict with MySpace's), but I haven't seen anything that says you can, either. Those terms can spell out how you'll be using their thumbnail picture and display name (and storing them) as part of how your service operates.