Hello skinniders. Welcome to the Apple Discussions!
1. What is the capacity of the Airport Extreme's router memory and how long does it save this memory for?
As far as I know, Apple has never published the amount of "memory" in any of their base stations. Obviously, it would be the amount necessary to store both the active configuration information, and up to 5 configuration profiles as a minimum.
2. Is there any way to add to this memory?
No. Apple doesn't offer either an Authorized Dealer install or a Do-It-Yourself memory upgrade option.
3. How does one reset the router memory?
You can reset the base station's memory, by performing a "hard" reset. For the AirPort Express Base Stations (AX), by either the "hard" or "factory default" reset methods.