Holders of valid U.S. and Canadian passports do not need a visa to enter Hungary but the passport should have at least 6 months left before it expires. Their desk is located in the lobby of the airport. Make sure it is metered. Check out Getting Around on this website.
Connections are available on many major European airlines.
Yes, the booths are open. Check Facts about Hungary on this website. Check Facts about Hungary on this site. Most of the stores will be closed. Hungary has a temperate climate.