Does this mean that if a crownholder does not issue an AR for a particular Crown during a 90 day period, then that Crown can issue 2 ARs in the next 90 day period? Or does it mean that if no ARs are issued at all from anybody, then the awards pool balance rolls over to the next 90 day period, instead of being reset to zero?

It's just a reference to the pool itself as for ARs, they are use it or lose it.

So ARs can be used completely independently of TRs? Does this mean you can choose to rent your TR to a mint, or sink it in the liquidity pool, but either way you still get to give out an AR as well?

Yes your AR is completely independent of your TR, unless your crown is indebt. Indebted crowns sink their TR into the liquidity pool to cool off and cannot issue ARs during the debt cycle.