I'm a level above an amatuer in this, having studied data assimilation in college (I was a Math major). Numerical errors are a given, errors in general too, actually, especially in the measurements used to train the hidden parameters (it is very similar to supervised ML). Data assimilation is a collection of techniques used to tackle that issue: 3dvar, 4dvar, kalman filter, extended kalman filter, ensemble kalman filter, particle filters, and so many others are used to find the most likely (minimum energy etc) hidden state (the mean and covariance) from a given set of measurements+associated covariances and then that resulting hidden state is used to run the model "forward" (generally in time). AIUI, DA was specifically developed for weather modeling. It is definitely true that weather modeling is a heavy employer in anycase. This is on top of the methods used to solve PDEs ie so that they are forward/backward stable etc.

The errors in the weather forecast are not the result of chaos: they are the result of the errors in the measurements (recorded in the observation covariance) and the sparsity of the measurements themselves vs the size of Earth, for example, and limitations in model resolution (consider a FEM grid over the entire surface of Earth). The effect of chaos just compounds these errors near bifurcations around fixed points.

Perturbations are not used in the way you think; think of Taylor series approximations around specific points of interest.

Anyway, I work on compilers/auto-vectorization now (lol), so I'll defer to The Expert, if such person wants to chime in.