1, 2, and 3 are all correct and mean the same thing.
3-1 is incorrect.
4 has what's called a "dangling" or misplaced modifier; it's incorrect.
5 is correct.
1. Walking in the park, I saw Mina.
2. When I walked in the park, I saw Mina.
3. When I was walking in the park, I saw Mina.
(Does #1 mean #2 or #3? Are they all the same?)
3-1. Being walking in the park, I saw Mina.(Is this one correct?)
4. With night coming on, we started for home.
5. Because night came on, we started for home.
(Does #4 mean #5?)
2 answers
Essentially they mean the same thing, but grammatically you should never start a sentence with the word because. Instead try using; as a result..., this caused..., or even the reasoning for this/ that is...