Exactly. Setting up a conventional gateway and merchant account takes weeks at best, complete with several contracts, credit check, authorization forms, and several rounds of changes to your website if you're taking payments online.
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.