Absolutely. We do that. Now, you need to think carefully about how you brand and control that experience - but there are several outsource providers that can be integrated to the point that either users absolutely have no idea it's someone else or just match your look. We went with the latter because frankly I'd rather not ask someone to trust me with their credit card information, I want them to trust someone who's whole business is around getting that right.
I'd worry about having a good looking process and having the best pipeline into it, and play up the credibility of how that's handled (and your flexibility). By outsourcing for example we were able to totally delegate handling currency conversion and taxation issues.
Of course. This is how a many companies operate.