I have to generate a "toy dataset" with one human chromosome and short reads mapping on these contigs.

Which one would you pick? I would go for chromosome 21 since: 1) it is short, so less data 2) it has no gender bias (depth) like X/Y chromosome.

Is this a good pick or would you advise to go for another one?

Are there pre-processed datasets anywhere with contigs and corresponding reads filtered for just the specific chromosome?
Depends. What sort of data? What's the goal? Chromosome 21 is pretty sparse, gene-wise, even taking its size into account. Chr19 is usually my go to, as it's still quite short while having a decent gene density. Really depends on what your goal is. Avoiding X/Y is probably a good idea, otherwise anything after chromosome 13 are all pretty small.