I'm curious about the use of the genitive here for židlí. Am I perhaps correct in thinking that kolik, similar to mnoho, takes a genitive object because of the likelihood that the number being spoken of is higher than four?
I think you are overcomplicating it with the likelyhood of more than four. It just requires it, that's enough. If any reason at all then I would think about the partitive genitive.
Note that quam or quantus alsonuses genitive in Latin.
Yes, it does not only work for numerals higher than four, but also for indefinite numerals (which is this case) that take the genitive plural.
Could we say: Kolik židli chcete? Is the original order more natural in Czech?
Both are fine but they are slightly different. Often they can be used in the exactly same situation though.
Kolik chcete židlí? When you are asking about the numbers of multiple kinds of things (first židlí, then stolů,...) you are more likely to use this one.