A 120 treadmill with 15 amps plugged into a home outlet with a 20 amp circuit. Is that the only thing that can be plugged into that circuit? When the treadmill is cut on the lamp in the same room on another wall will dim for a second. Is this a problem?

You have 5 amps left if that's the only thing running. A lamp requires an amp or less (60 watts requires half an amp). The lamp dims because the voltage drops a bit -- especially when the treadmill starts up (it requires extra power for that). It's not a problem.

All things with motors take a lot of power when the start up