You do not need (and cannot get) a short-stay visa for the Schengen area (including France) as the US is on the list in annex II of Regulation 539/2001. You can therefore present yourself at the border (i.e. take a plane to Europe) without any prior formality.
Theoretically, you might be asked to show you fulfil a number of conditions like having sufficient financial means (e.g. that you have a credit card) or a valid purpose for your stay but in practice and unless you raise some red flags, you will just be asked a couple of questions and stamped in.
And yes, you can go in and out of France as much as you wish for the next six months (or even after that), provided you never exceed 90 days in any 180-day period. Since you do not have a visa, there is no need to renew anything. You will merely get entry and exit stamps that document your stays in the Schengen area and are used to enforce the 90-day limit I just mentioned.
It's not relevant for you as US citizen but for people who do need a Schengen visa, what is allowed depends entirely on the visa. A Schengen visa can be valid for up to 5 years but can also be (much) more restrictive, there is absolutely no guarantees that it will be valid for six months or allow a 90-day stay and no direct link with the 90/180 days rule (which, for visa holders, is just an upper bound on what can be granted).
First-time applicants usually get a visa that covers their first intended trip, nothing more, and might be forced to apply for a new visa for each subsequent trip (which means providing all the documents to the consulate, paying the fee and waiting for the decision again). It's only after several trips and if they have a good reason to visit the Schengen area regularly that visa nationals get a multiple-entry visa valid for one year or more (in which case the 90-day limit applies).