1) Power gain for Oscillation means the transistor draws DC power to amplify the AC signal to boost its output power with some sort of 2nd order effect such as discrete or parasitic or distributed conductor LC values. THe LC values provide the impedance ratio and thus voltage or current filtering and phase shift to cause oscillations.
The "Barkhausen" criteria for oscillation is positive AC feedback and gain =>1. AC Analysis of all circuit, cable and component LC values are necessary to determine if the impedance ratio and voltage gain leads to ringing or steady oscillation or a growing oscillation to saturation ( square wave).
Note that there are also RC oscillators which are 1st order filters but with hysteresis on negative feedback which will create a "relaxation oscillator" .
This uses negative DC feedback ( to become self DC-biased) with hysteresis causes a time delay with a shunt capacitance and thus a 90 deg phase shift with each half cycle netting an equivalent positive AC feedback of 360 deg. In this case the input appears as a triangle wave and the output a square wave, yet the same principles apply drawing DC power gain with filtering to produce AC oscillations with more output power than the input. In THIS case the input is a DC offset which causes a slew rate which contains enough signal of the frequency of oscillation to grow to a steady saturated oscillation very quickly, often symmetrical about V+/2.
When unity gain with 0 or 360 deg feedback (positive) the output is sinusoidal. We know emitter followers provide unity voltage gain and power gain from current amplification. But when driving a series inductive thin wire with a very low driving impedance a high Q ringing on square edge signals will cause ringing from impedance mismatch and in some cases degenerative feedback will sustain the oscillation at very high frequency due to the emitter LC cable loading effects.
Other examples are feedback with gain with LC parallel resonance are Colpitts Hartley and Crystal oscillators which are emulated by LC components to net a 0 or 360 deg. etc. phase shift at resonant frequency.
THe startup time is by the dampening factor or Q or real/reactive impedance ratios of the circuit. ( where I will stop here ,as I fear I have said too much already). Crystals often have Q=10k while LC circuits ~100 max from physical constraints and RC hysteresis Oscillators limited only by the Gain-BW of the amplifier.
2) All transistors and diodes are call Active devices due to the semiconductive slope of I/V which gives rise to gain of voltage or current depending on the load with a DC source. Passive devices (RLC) may also have an AC voltage gain or current gain from a DC power step voltage from impedance ratios of a tuned circuit but are not able to sustain the oscillations because there is NO ACTIVE SEMICONDUCTOR device to provide POWER GAIN. So V or I gain is always at the expense of a rise in output resistance for passive only circuits. THus an Active device is necessary for Power gain. THe power gain is limited by the size of the device, where efficiency of AC power gain and DC consumption depends greatly on the circuit design with tradeoffs for linearity, distortion etc.
THe Reactive impedance is used to determine if there is gain relative to the the resistive loading. Dampening factor is derived from this impedance ratio.