The 'tiny' RC helicopter motor might draw a TON more power than another random motor. The battery might be able to supply it OK, but after a few seconds the battery becomes -unable- to supply enough power, so the voltage drops and the Arduino resets.
The motors that seem to work, run one up to speed, grab the motor shaft to stall it, an I'd bet it resets the Arduino.
Measure both with a multimeter ammeter (in 1A or higher mode.)
Regardless of anything else, yeah, it's NOT a good idea to run 'power' items off the same source that powers the controller. The Arduino power system is designed to power itself, and the -circuit- part of plug-in/shield/accessory boards, but not power devices such as motors/solenoids/servos.