You have to use “des” when there are many objects (lapin**S** here)

and “de” when there is only one object (1 lapin here)

Sentences 2 and 4 are right.

In this context, I think that ‘de’ is only used in a negation and when you talk about quantity.

Je ne vois pas de lapins.

or

Je vois beaucoup de lapins

but not with numbers!

Je vois deux lapins

Note: A negation is not with ‘de’ when you use the verb ‘être’. Then it’s just

Je ne suis pas un lapin.

This problem is probably one of the worse…
It has a lot of exceptions and cases to handle.

For exemple you could say :

• J’ai des lapins.

But you can also say :

• J’ai beaucoup de lapins.

But, this also works :

• Je n’ai pas de lapins.

And this also :

• J’ai de très gros lapins.(1)

While this also works..

• J’ai des très gros lapins.(2)

Yes I just wrote a sentence where both de and des fit, but actually doesn’t have the same exact meaning. (1) uses de to increase the impact of the adjective très gros, while (2) uses des just as a plain article. It would be actually quite hard to explain the exact reason why de or why des on a lot of cases (as a French Canadian) since often we use the one that just “Sounds better”.

There are a lot of specific and different cases.

I would recommand you this website that I found searching on google that explains clearly when to use de, des, du, etc.

EDIT : I also wanted to add that, in many cases like these listed, des is used for a little groups, while de is used for big groups.

• J’ai des lapins. (some rabbits)

• J’ai beaucoup de lapins. (a lot of rabbits)

De is used instead of des if you are using an adjective placed before the noun.eg: “des oiseaux blancs” but “De grands oiseaux blancs”

These are correct…

• J’ai des lapins : standard usage

• J’ai beaucoup de lapins : des reduced to de is standard when used with collective nouns (beaucoup de, un groupe de, une quantité de, une grappe de, …)

• Je n’ai pas de lapins : des reduced to de is standard when the verb is negative
• J’ai de gros lapins : des reduced to de is standard when the noun is preceded by an adjective