Why would one use the genitive case here and not accusative? I think if I read this sentence in English I would say that "many small children" is accusative, so "mají mnoho malé dětí" similar to "I have coffee" or "Mám kávu". However, if I'd say that "I have many of the collectible cards", so here collectible cards is a subset of all of the collectible cards, I would use genitive. Are there any rules when to use genitive and not accusative with the verb mít?
jorubsi, it is not the verb but the word 'mnoho' that is the source of your confusion here. Unlike the word 'many' in English, the word 'mnoho' in Czech is not an adjective. It is an adverb. So the sentence is literally something like "They have muchly of small children."
Or you can think of the sentence as "They have a lot of small children," in which the English sentence too is followed by a genitive phrase beginning with "of".
(In the latter example, some would consider 'a lot' a noun phrase and others would call it an adverbial phrase. Either way it is followed by a genitive.)
Thanks for asking, I wouldn’t have picked up why. This is from the explanation under the skill page: …the genitive. It is used for objects of many verbs and with a few prepositions, and also occurs in constructions with nouns/noun phrases (often to show ownership), adverbs of quantity…
The word MNOHO is your culprit. It is considered a number (kind of) and behaves as one. And Czech numbers are really really weird. 2, 3 and 4 behave one way and everything above 5 including generic quantity descriptions like "many, few, more" etc. behave differently.
Read more on numbers under the lessons for numbers. 5 and up are described under Numbers2