The answer to both is no.
Taxing authorities decide how much money the property tax has to raise each year, say $1 million. Assessors estimate the total assessed value of all taxable property, say $100 million. A tax rate is calculated by dividing the amount of tax to be raised by the total assessed value: $1 million divided by $100 million equals 1%
If your home's assessed value is $100,000, your tax bill will be: 0.01 times $100,000 equals $1,000
If total assessed value doubles to $200 million, and the amount to be raised stays the same, the tax rate will be: $1 million divided by $200 million equals 0.5%
Your taxes, if your home doubles in value, will still be $1,000: 0.005 times $200,000 equals $1,000
If assessed value increases, and the tax rate remains the same, taxes will rise. The taxing authorities are demanding more money, even though they have not changed the rate. 0.01 times $200,000 equals $2,000
The Assessor's office does not control the tax rate.