Is there an experiment which can test whether the Planck length really is the shortest possible length?
It's more accurate to talk about Planck scale and not think it as strict limit.
Planck scale is where the structure of spacetime itself becomes dominated by quantum effects and the structure of spacetime may start looking really strange. Strange theories like loop quantum gravity, causal sets, causal dynamical triangulation, fractal cosmology, etc. work at Planck scale. According to fractal cosmology spacetime is 2-dimensional in Planck scale and gradually becomes 4-dimensional in larger scales. Loop quantum gravity sees it as "foamy".
Planck scale is also the area where measuring distances (differentiating with different positions in space) becomes impossible.
Which suggests internal structure of some kind.
I'm wary of any explanation that says "Well, it's just random", because randomness turns out to be a complicated process.
It's hard to imagine that some kind of prototypical base quantum would be inherently random just because.
I suppose it's possible. But it would be unexpected.
These units are very useful in QM and as "constants" (not the best choice of words since Planck units intentionally ignore constant values to some extent) that can be used to describe a relativistic universe.
*You can use Planck Length/Time to put a lower limit on any possible wavelength that below it a wave cannot exist, other Planck units can also be in the ballpark of the smallest possible unit/quanta of various things in various theories.
The uncertainty principle states that the product of the standard deviation of the momentum and posisition of a particle is bounded from below by h/4pi, where h is the plank constant.
This means that, in order to probe small distances, we must have great uncertainty in the momentum invovled, which means that there must be a lot of kinetic energy in the system. As the distances we probe become smaller, we have an increasing amount of energy in a decreasing amount of space.
E=mc^2 tells us that energy and mass are interchangeable. Specifically, energy, like mass, causes gravity. In order to probe below the plank length, we would need to put so much gravity in so small a space that we would form a black hole. However, because we cannot observe the inside of a blackhole, this prevents us from observing anything smaller than the plank length.