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.

It's flat out wrong, you can go into Airport Utility, check the connected clients list and see the connection speed and quality of all devices which will vary greatly.

The only time slowest common denominator comes into effect is when you exit the router to another network (typically the internet), and even a poor connection to your airport is likely to be faster than your internet connection - an old fashioned Wireless G connection to your router from say an old iPhone 3G might run at up to 54Mbps, and unless you are on a Fibre net connection this will likely be faster than the bottleneck which is your net speed.

One time when it might come into play internally to your network is if your machines ever talk to each other rather then as most people do and just use the router to hit the internet. If you have a device running at 54Mbps (say an old Windows netbook) and it's doing a file transfer to your new iMac running at 1Gbps, then obviously the speed is limited by the slowest device, but only for that specific piece of network traffic and it cannot affect anything else.

According to

http://en.wikipedia.org/wiki/IEEE_802.11ac

one of the new things in 802.11ac is "Coexistence mechanisms for 20/40/80/160 MHz channels, 11ac and 11a/n devices". But I have no ac device to try it out on.