If I place an order to buy 1 share for $1 and another share for $1.01 because the market data shows that there is only one share available for $1, then the exchange needs to process my $1 order before my $1.01 order. More complicated scenarios exist, too, especially once you take into account resting orders and modifications to existing resting orders.

Right, unless exchange explicitly documents the fact that order of fulfillment is not guaranteed if you place them within the same time interval.