If my school district says it wants to extend an existing bond and not increase the tax rate and the assessed value of my home never increases/decreases(hypothetical), will my taxes stay the same?
If you refinance your mortgage, does your mortgage payment have to go up? No.
Refinancing a bond works the same way.
If the assessed home value increases,
Your home value might increase while others decrease. If the county-wide valuation increases, the school board will get more money.
is the school district paying down the bond faster?
Is there a property tax "slice" dedicated to that bond? If so, they might pay it off faster, or they might accumulate the extra money in a rainy day fund in case property tax receipts drop in the future.
You'll have to ask them.
Or only taking a fixed amount according to a repayment schedule?
In my county, each dedicated bit of the property tax pie is a a percentage. When property taxes go up, the pie gets bigger.
YMMV but I doubt it. In any case, your county should have a web site explaining all this.