The absolute instability is a subject of considerable physics interest as well as a major source of self-oscillations in the gyrotron traveling-wave amplifier (gyro-TWT). We present a theoretical study of the absolute instabilities in a TE01 mode, fundamental cyclotron harmonic gyro-TWT with distributed wall losses. In this high-order-mode circuit, absolute instabilities arise in a variety of ways, including overdrive of the operating mode, fundamental cyclotron harmonic interactions with lower-order modes, and second cyclotron harmonic interaction with a higher-order mode. The distributed losses, on the other hand, provide an effective means for their stabilization. The combined configuration thus allows a rich display of absolute instability behavior together with the demonstration of its control. We begin with a study of the field profiles of absolute instabilities, which exhibit a range of characteristics depending in large measure upon the sign and magnitude of the synchronous value of the propagation constant. These profiles in turn explain the sensitivity of oscillation thresholds to the beam and circuit parameters. A general recipe for oscillation stabilization has resulted from these studies and its significance to the current TE01-mode, 94-GHz gyro-TWT experiment at UC Davis is discussed.