Do all computers/devices connected to an airport extreme run at the same speed? My mom just bought an Airport Extreme router recently, and she said the salesperson at the Apple Store told her that all devices on a network run at the same connection speed, and that speed is determined by the slowest device on the network. So if you have a shiny new Mac that has the potentential to run at 1 Gbit/second, you'll be slowed down to a 10 Mbit/second speed, just because you have your old phone connected to the wifi, downloading some apps at the same time. This doesn't really make any sense to me. 

(Please don't speculate, I can do that myself. Answer if you have experience with this topic).
No - this is demonstrably false for Airport hardware even before the latest generation where 802.11ac is introduced.

Simply connect two or three Mac at different distances (1m, 5m, 10m) and run wireless diagnostic on each.  You will see different transmit rates and MCS index for each device which correlates to how fast the radio link between each device and the base station is based on current interference and signal to noise ratios.