Processing and feature extraction from very short signals
Though problem! However, signal processing might have a tool at hand: It's called compressive sensing, and reduces the number of samples you need to sub-Nyquist-rate levels.

It's a bit nonsensical to derive the math behind compressive sensing for signals that are a sum of sinusoids (that being the only thing that I could do without literature) if that doesn't describe your signals overly well. However, the idea is:

If you want to know about everything happening with less than a given frequency over a specific time, you'll need all of the samples at a constant rate (typically, twice the highest frequency for real-valued samples). Then you can, for example, do a base transform to get the samples from time domain to frequency domain using the DFT.

For the following, assume that you could, if you had the "full, long, evenly-sampled" signal $\mathbf S \in \mathbb R^N$, define a base transform $\mathbf T$ to transform the time samples vector to a result vector giving you the information you want $\mathbf Y = \mathbb R^N$. That base transform would then simply be a matrix:

$$ \mathbf Y = \mathbf S^T \mathbf T$$

If what you want is sparse, ie. the dimensionality of the result vector would be much smaller than the observed samples vector, the transform matrix wouldn't be a square full-rank base transform, but a matrix that would have $N$ rows, but fewer columns.

The point here is that most of the values in $\mathbf T$ would be pretty small -- setting those to zero will incur an error, that "feels" like noise, but given a nonzero upper limit for allowable distortion, you can find a sparse matrix that still can allow for reliable conclusions about $\mathbf Y$.