It is tricky. The keres = to look for and kér = I ask for (politely) are considered different words (it may be useful to remember them together, being very similar)

the -és/-ás suffix means to turn a verb into a noun. So kérés = a request. The -es suffix is also used in a different way, but not on verbs. So keres is a root by itself, not derived from another word.

That's a bit very colloquial, even dismissive. Hungarian has its own colourful words to describe policemen, like zsaru. Rendőr is the official term.

Google Translate can be helpful. 'Cop' translated to 'zsaru', not a word I am familiar with. In my experience every language has slang words to use for the police. I consider cop a slang word even though it is very common and everyone understands it.