Are there any successful open source based startups that have inspired you?
Building around someone else's APIs isn't such a cardinal sin as you're implying. The big questions is what are you building? Owning and controlling your full stack is important if you're building something huge, but you can build something profitable leveraging someone else's APIs. There's a bias here towards building BIG businesses rather than lifestyle businesses, so advice about avoiding third-party APIs would come from that. If it works for the project you're doing, then do it. If not, then don't.
Consider avoiding GPL if you are going to make modifications, want to redistribute your work, and cannot support a process for releasing that work.
Nearly all permissives licenses are good (Apache, MIT, most BSD, etc).
Nearly all licenses allow you to make modifications, it's just a question of whether you are re-distributing the work covered by the original license or not. That's when things get interesting and you should check the licenses carefully in advanced.
But if you have no plans to redistribute then nearly all permissive licenses are good.
BTW: This is a 30-second rule I use... it's not fact, IANAL.
In short, building around a third-party open source project is probably an easier win than building around a project that you founded and maintain. (Forking, on the other hand, is not something I'd recommend.) While not required, it's a really good idea to make friends with the community organizers and, ideally, find some way to contribute both code and funds upstream when you become successful - after all, your business is based on their continued existence.
That said, although it's certainly much harder to succeed under these terms, maintaining your own project is a valuable, rewarding experience, and gives you significantly more control over its direction.
Just remember that open source code has an accounting value of 0 - so any open source IP you create is not capturing value in the company, and unless you're careful, one-to-one professional services don't either. You'll have to figure out how to make your company valuable with valueless code another way. (Hosted services are a good example.)
Avoid Affero GPL (AGPL). Everything else is fine for commercial (closed close) startup based on open source software.
http://finance.yahoo.com/echarts?s=RHT+Interactive#chart1:sy...
That's about as good a 5 year chart as you'll see in today's stock market. They represent one particular business model, and they've done exceptionally well, but they're just one example of a company that's taken open source all the way to the bank.