You can’t get a tourist visa after entering, but as a US passport holder you would get a free 30 day visa exempt stamp and could extend that once for 30 days for 1,900 baht.
Depends on your passport country. Some require a visa to enter. Many get what’s called a visa exempt entry for 30 days, which can be extended in country once for 30 days.
There isn’t a limit. However it depends on where you apply for it. Some consulates will only issue one per year from that consulate. Others may impose their own rules saying you can’t have more than X in a year.