Yeah, but not with PayPal's "Website Payments Pro". It does usually take 2-3 days, but it's really quite painless. It certainly doesn't require a credit check.
PayPal probably already has an App too, so all they need is to give you is a reader.
Square is operating online too. That's what the mobile internet-connected computer is, that their reader is attached to. PayPal App + iPhone + reader = Square except cheaper, more reliable, and probably accessible to more people.
PayPal was revolutionary. The only thing I see here from Square that's even kind of new is the reader component, and that is hardly novel.
You're right that they do take an SSN. I always assumed that was for tax or risk purposes. They don't say they do a credit check, but you're right, it is possible they do.
PayPal probably already has an App too, so all they need is to give you is a reader.