> It gets adjusted nearly every year for most people.
For inflation, yes. Real re-valuations that mark the house to market are a lot more rare, and tend to only happen when a house is sold.
> No, you can't. Not really.
I have personally done this a couple of times, and all it tends to take is a nice letter explaining why the value is too high. At worst you need to get a new appraisal to support the letter. The value is rarely too high, though.
Property taxes in my area are also relative: the tax rate floats depending on how much the local government needs to raise. The sum of the appraisals for the city is added up, and your property tax is your proportional share of the total amount that needs to be raised. For me, that's usually around 1-1.5%.
This is all to say that property tax is not at all a tax on unrealized gains of your house. It's a tax on an imaginary number that occasionally adjusts to be similar to your unrealized gains, and often even has a rate that floats relative to it.