In my Slayer Exciter circuit, the current from the battery flows into the primary coil (with four turns) and therefore, a current is induced in the opposite direction in my secondary coil. This current flows into the wire coming from the base of the transistor and opposes the current flowing through the 22k resistor. Is there a specific name for this?
Without it there's no oscillation - the circuit will find a steady state where constant current flows through the coil. By affecting the primary switch by the voltage on the secondary coil, you provide a self-sustaining oscillation - off state causes on state and on state causes off state.
The triode (vacuum tube) is 'on' with zero gate-to-cathode voltage so it does not need the bias resistor to turn it on. There is a hidden current path from the high voltage end of the secondary to ground in your circuit (through arcing or corona discharge).