As a US passport holder, you're "visa-exempt" and will generally be granted 90 days on arrival, no questions asked:
The nationals of the following countries are eligible for the visa
exemption program, which permits a duration of stay up to 90 days: ...
Now, making a quick visit to another country for the sole purpose of renewing your visa is known as a "visa run". Most countries will get suspicious if you do this too often, but fortunately, anecdotal evidence for Taiwan seems to indicate that they don't care:
According to experiences on this forum, you can continue to do this as
long as you like. Of course, like anything, Immigration has the final
say, but as long as you keep your nose clean (ie. don't overstay your
visa, don't commit any felonies, etc.) you should be fine.
(courtesy "Steve4nLanguage" on Forumosa, the definitive Taiwan forum)
So the answers to your questions appear to be:
All that said, I wouldn't rely on this for more than a few renewals. If you're planning on staying in Taiwan for a longer time, you'd definitely best be off working out some sort of "real" visa that actually allows you to work legally.