Native speaker here. Well, the question about using на, за and для is a bit complicated. So I'll try to explain. At first let's see with which cases we can use на, за and для. Let's start with the last one. Для takes genitive and that's all. На takes either accusative or prepositional/locative. And за takes either accusative or instrumental. Now bad news. These prepositions have many different meanings including the one of purpose/use. Let's begin with для: 1. something for someone: entrance for spectators - вход для зрителей; 2. something for something: folder for papers - папка для бумаг; 3. do something for something/someone: I am doing it for him - Я делаю это для него; 4. do not know how to explain but the example will help, I think: It was unexpected for me - Это было неожиданно для меня. Now, the next one is на. It has 10 or more meanings, I think. So let's just see how it works: На indicates the intended purpose of the object or action. So it means that на is more or less the same as для in this case. But... When indicating the goal towards which the action is directed, the prepositions differ: 1) semantic shades (для brings a greater shade of purposefulness): использовать для местных нужд - use for local needs; истратить на местные нужды - spend on local needs; 2) stylistic shades (на bring a colloquial shade): Для чего вам эти вещи? — На что вам эти вещи? - Why do you need these things? And btw на takes accusative in these examples. So, what do we have in the example время на кошек? The meaning is that maybe someone will spend some time on being with cats (and possibly playing with them). So, to spend time on something - тратить время на что-то. And then: у тебя есть время, и ты можешь потратить его на + accusative ~ у тебя есть время на + accusative. Finally за. За has more than 20 meanings. In "Я плачу деньги за ремонт" за indicates an object, circumstance, etc., which is the reason for some actions. And if we speak about purpose using за... Well, the easiest example i can think of is this: fighting for justice - борьба за справедливость.
время will be the nominative-- 'time to you is'. The unexpected forms in other cases are the result of historical language development, in a very general way like how мать has unexpected forms in the other cases-- it's irregular. кошек is functionally accusitive, so, as an animate plural noun, it takes genitive forms, like ph516503 says. Genitive plurals have a lot of different formulas, but actually, as a feminine noun ending in "a", кошка's change to кошек is pretty common/regular.
I'm ready to be corrected by someone who knows what they are talking about :-) , but I think these are the genitive plurals of each noun. Personally I haven't done them yet, but I keep coming across dark hints that genitive plurals are very complex and will be tackled later in the tree...