...I'm still not sure if the visa is a document attached to the passport or as stamp on a visa page.
Your "visa" is permanently affixed to one of the pages of your US Passport by the Consulate
. When we applied for the Visa, the Consulate required that we leave our US Passport with them until they'd granted or rejected our request for the Visa.
So what would happen if I have a visa with a certain arrival date but I actually arrive earlier?...
As far as I know, there's no arrival date on one's Visa. There is a date, if memory serves, on which the Visa was granted. During our interview, we were advised to be prepared to go to France within a few weeks of the Visa being granted (3 weeks comes to mind...but I don't really remember for sure). It would be useful to ask the Consulate if there's a required or recommended amount of time to move to France.
If the Consulate still keeps one's US Passport and you wanted to go to France before receiving your Visa, you'd need to request it back in order to travel internationally. If they return it to you before the Visa application is processed, you'd have it in hand, of course. In either case, though, you would have to return to the US to the French Consulate to have the Visa attached to your US Passport.
As a result, it doesn't seem possible you'd arrive in France "early"...you'd simply arrive as a tourist who'd have to return to the US.
If the Consultate still retains one's US Passport while deciding whether to accept or decline one's request, I strongly recommend you ask the Consultate whether requesting your Passport back before they've concluded their deliberations will void your request for a Visa.