Will an electric heater coil stop drawing power when the surrounding temp is the same as the coil's temp?
The heater will never stop drawing current, the current will be quite constant because it is made of material with low thermal coefficients of resistance.

The heating element loses heat by three principles: conduction (contact with cold air), convection (wind) and radiation (far-IR)

Conservation of energy works also in this case. When heating up, consumed power is used to heat the element, and cover heat losses.

The hotter it gets, the higher percentage of input power it dissipate into air.

When it reaches its maximum temperature, it can not heat itself more because all input power is lost in air.

Side note: some heating elements can get hotter then their own melting point (e.g. clothes iron) and can not work without temperature controller.