Is the electron constantly acting with some inherent energy "dancing around" and that this dancing creates randomness that is simply to small to measure and thus leads to the electron having a probability-like nature?
Before measurement, the electron has no definite position or momentum, so you can't talk about trajectories or dancing around. There really is a probabilistic nature to the measurements, but it is not due to not knowing enough about the system like it is for something like coin flips or rolls of dice. When you try to search for "deeper things" causing the probabilities due to limited knowledge, you find that there is none. It really does seem like it is an inherent part of nature.
That the Heisenberg uncertainty principle is not describing our limited ability to measure the particle's position and momentum but that the electron itself is probability-based?
The HUP is not directly due to the probabilitic nature of measurements of QM systems. The HUP relates simultaneous measurements of position and momentum, but even measurements of single observables have probabilities associated with them. The HUP only deals with the uncertainty of two specific measurements. In other words, the HUP only characterizes the probabilistic nature of measurements of QM systems, it doesn't define it.