Understanding facts about regular languages, finite sets and subsets of regular languages

Question

I am aware of following two facts related to two concepts: regular languages and finite sets:

• Regular languages are not closed under subset and proper subset operations.
• It is decidable whether given regular language is finite or not.

However I feel these facts are quite insufficient to prove whether following statements are true or false:

1. If all proper subsets of $L$ are regular, then $L$ is regular.
2. If all finite subsets of $L$ are regular, then $L$ is regular.
3. If a proper subset of $L$ is not regular, then $L$ is not regular.
4. Subsets of finite sets are always regular.

I feel 3 is false, as regular languages are not closed under proper subset operation. Right?

But I am unsure of the other points. What facts I am missing to answer above statements true or false?