"The tourist is taking few pictures."
Translation:Keveset fényképez a turista.
That is one of the several ways a verb is created from a noun. It will usually mean "doing the normal, regular activity with the given object". There is probably no specific rule, you just have to learn them one by one. But you will see many verbs that are a derivative of a noun this way.
I would say "egy keveset" for a "few" I personally don't think it has the same meaning in the solution. "Keveset" on its own rather means in this case there was a requirement earlier, and compare to that, the result is less. I would say "Néhányat fényképez" or "Egy keveset fényképez". But it is my opinion.
Yes, it is common, no, I don't know about "rules" but I suppose it's more common for short vowel+long vowel 2 syllable words than the opposite. A lot of words simply have two stems that have to be learnt. The way these stems differ is quite regular (especially if you have a basic idea about old Hungarian phonology and word structure). Btw watch out, it's kevés and not kerés and kereset is like earning, related to keres the verb.
Oh and one thing I forgot to add back then but it might be worth mentioning - changing stems (apart from the a, e thing at the end like medve→medvét, macska→macskát) aren't a productive feature of the language - that is, there is a finite number of such words and newer words are highly unlikely to adopt a pattern like that. So they may be quite numerous but still a finite list of exceptions.