Cats and their aversion to water

So, today as I was watching the cat try and jump through the snow without getting any on him, I was feeling rather bad for him. He so hates getting water on himself and that includes frozen water. I was wishing that he did not have to be outside but I am caught in a quandry.
1) This is technically the neighbour's outside cat that they don't let in but I do cause he cries at my window when he wants inside. There are days when he does not want outside at all but...
2) Can I keep the neighbours cat inside?
3)If I keep him inside then I need a litter box and where in the world would I keep that?
4) What kind of litter box would I get to fill with what kind of litter?

So many pressing questions and so little expertise to answer them.
So what do you think? Comments?