Hi Gasant,
1) Does the transformer supply the Drain leg of Fet with power
The drain is just like a switch controlled by the gate and the supply is coming from the transformer.
2) Does the Drain leg of the Fet make the run dc circuit.
Run dc ac signal source is from one of the output of the secondary winding in primary side. The primary winding have to be energized in order to get the ac signal source from the secondary winding.