Do US citizens really get a year visa-free in Georgia, and how many days for Tbilisi plus a Kazbegi side trip?

Yes - US passports (and most Western ones) get up to 365 days visa-free, stamped in at the border with no fee. For Tbilisi plus Kazbegi, 5-6 days is comfortable: 2-3 in the city, then 1-2 nights in Stepantsminda reached by shared taxi or marshrutka from Didube station. The drive up the Georgian Military Road is half the experience, so grab a window seat on the right side going up.