1. He is a very kind boy.

Question

2. A fox is a cunning animal.

(Is the meaning of 'a' before 'very' the same as the meaning of 'a' before 'cunning'?)

3. Once there lived an old man.
4. Once there lived one old man.
(In this sentence does 'an' mean 'one'? Are both the same?)

Answer 1

1 and 2 -- Yes, "a" indicates there's one.

3 and 4 are about the same, but 3 is smoother.

Answer 2

The "a" is a modifier of the nouns - boy and animal...

"an" means "one".

Answer 3

"an" is used instead of "a" when the following word starts with a vowel, such as "old man"

Answer 4

It might help a little more to distinguish between the use of "a" and "the."

It is a correct answer.
It is the correct answer.

In this case, "a" infers it is one of other correct answers. In contrast, "the" indicates that it is the only correct answer.

I hope this helps a little more. Thanks for asking.