The difference is indeed the number. For the verb нравиться, the grammatical subject is the thing that someone likes. And the person who gets that warm feeling is in the Dative case:
- Мне нравится эта кошка. (кошка is singular, use the 3rd person sg. form)
- Мне нравятся эти кошки. (кошки is plural, use the 3rd person pl. form)
- Я всем нравлюсь. ("я" is singular, use the 1st person sg. form)
- Вы мне нравитесь. ("вы" is plural, use the 2nd person pl. form)
As you can see, нравиться has all 6 personal forms, like other verbs. It is just that you only use forms for the 1st and 2nd person when SOMEONE likes "me", "us" or "you".
Sure. It is кошки.
If you get confused, remember than Russian "нравиться" works similar to the English "to seem" or "to belong". An object SEEMS such-and-such TO A PERSON who talks about it. Technically, it is the person's brain that perceives the object, but you still talk about it as if the object is doing this to you.
Later in the course you'll notice a number of Dative expressions that mean some state, emotion or possibility. In English this is not really popular.
No, for two reasons.
- The nominative case is used with что за, so it would be "что за кошка". Of course this also means that you need to use singular нравится.
- This is not an appropriate usage of что за. It may be grammatically correct (I think, don't take my word for it), but the Russian 'что за' does not directly translate as 'what kind of'. It can often be used for 'what kind of', but sometimes it makes little sense, like in this case. I'm sure there is a more clear technical/rule-based explanation for this which I can't quite put my finger on.