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I've been taught that у and к both translate to "at" (as in chez in French).
"У меня есть книга" = "At me, there's a book" = "I have a book"
"Я к врачу иду." = "I go at the doctor"/"Je vais chez le docteur" = "I'm going to the doctor."
But then, how do we choose? I came across this sentence:
"Я не хочу жить у вас"
and I wondered why we could not say "к вам" instead.
Are there specific rules for у/к?
You've been taught wrong. К means "to" or "towards".
A-ha-ha, I recall a French comedy calls "Viens chez moi, j'habite chez une copine" so that gives an idea why this question comes to your mind at all. Once again, never made assumptions on how something is supposed to be structured in one language just because it's the way it's structured in some other language.
Even closely-related languages can approach the very basic concepts in a very different grammatical way. See what happens here, the French "chez" corresponds in some cases to "to", and in Russian in some cases English "to" corresponds to "у" and in some cases to "к" - and you make a false assumption that from that follows that phrase "я не хочу жить к вам" can have sense - but it just does not.
Languages do not function this way. There's no sense in your question. Well, there is some in that sense that one can speculate on how things turned out this particular way providing some comparative linguistic context, but practically - finding analogies in your native language when learning new one is a slippery slope.
Unfortunately, you just need to learn what verbs are used with what preposition.
The choice depends on semantics and semantics are to be defined by a case + preposition.
Hence, when we speak about a non-motional state, we usew y (= près de, au côté de; proche de; chez; voisinant) we say :
У меня есть/имеется вопрос.
Я живу у друзей.
Ключи - у соседей.
У кабинета врача - очередь.
When we speak about motion or actions addressed to other persons, we use к (à; pour; vers; chez):
У меня к тебе есть/имеется вопрос.
Я живу у друзей, к ним ходят соседи.
Ключи - у соседей, а эти ключи к замку не подходят.
У кабинета - очередь к врачу.
Strictly speaking, к in is more like à, not like chez (which in French stands both for a state and a motion/direction towards somebody's place).
