No, they aren't. Prop 13 limits maximum property tax rates to very low amounts as well as limiting assessment increases, so no jurisdiction is able to make up for artificially low assessments by jacking up the nominal rate.
> (as are the number if different fees and things like sales tax).
Mostly, it's state income tax, though state and local sales tax are also affected (there are general and program-specific revenue sharing mechanisms by which state revenue goes into local coffers to pay for local programs.)
Lower housing stock leads to inflated home prices, causing higher taxes paid by more-recently-purchased homes.
Actually, it's reassessed annually, there is just a limit to annual assessment increases except with qualifying events.
Which doesn't change the fact that Prop 13 limited nominal rates to lower values than many jurisdictions (including core Bay Area ones) were before Prop 13; jurisdictions aren't making up with artificially low assessments on long-held properties with higher taxes on other properties. Yes, those other properties are assessed closer to full value, but it's not offsetting anything. If you cut the food you eat on weekdays by 50% and that on weekends by 66%, you aren't making up for the reduced weekend calories by eating more during the week.
It is reassessed at market value after a non-family sale.