Any ideas on how large the range is on the AirPort Extreme
As you think about it, any range "measurements" that you might see depend entirely on how the measurements were taken. There is really no way that any range measurement...even if it is published...can predict the wireless performance when multiple obstructions like walls and ceilings are involved.
Even if you know the number of walls and/or ceilings, you have to know the exact type of construction that was used on the home to make a good guess as to how much of the signal is being absorbed.
A very general guideline assuming that the walls are standard sheet rock and 2x4s is that roughly 15-20% of the wireless signal, on average will be absorbed by each wall. Ceilings absorb more because they are thicker. It sounds simplistic, but the bottom line is that you need to minimize obstructions in the signal path for wireless to work
reliably.
and ideas of how I can improve my reception downstairs?
The "best" solution is to always run an ethernet cable from the main router to another router located in the area where you need more wireless coverage. This allows you to provide a strong wireless signal exactly where it is needed, providing the absolute best bandwidth (speed) on the network.
The "next best" solution...if you can't run the ethernet cable...is to use a pair of ethernet powerline adapters to transmit a pseudo ethernet signal over the existing AC power lines in your home. One adapter is located near your main router. The other adapter would be installed in the same room as the devices that need more wireless coverage. You would connect another AirPort Extreme or AirPort Express to the powerline adapter to provide more wireless coverage in the specific area where it is needed.
The last...and least desirable...option in terms of wireless performance is to establish a "wireless only" setup and use another AirPort Extreme or AirPort Express located approximately at a mid point between your main router and the area that needs more wireless coverage. The AirPort Extreme or AirPort Express would be configured to "extend" the wireless from your main router.
As you might imagine, this works fine if you are lucky enough to have close to a line-of-sight situation between the main router, extending device, and distant devices that need more wireless signal. But, if you are dealing with multiple obstructions, things get very "iffy" very quickly on these types of installations.
Xbox, Wii, etc. require as much bandwidth as possible...far more than a computer requires for simple internet browsing or email. The decision is yours.
