* Is it Ok to say ~ While I walking instead of "while I WAS walking"?
No, it is not OK to say "While I walking." Unfortunately, it is totally ungrammatical. You can say "While walking" instead of "While I was walking"; however, if you do, you need to make sure that the subject of the main clause is the same as the understood subject of the reduced clause "While walking." Since the trees weren't walking, you would need to revise the main clause. You could say: "While walking late at night, I thought that the trees looked spooky."
* Does it have anything to do with "fused participle"?
No, it has nothing to do with the "fused participle," which is the name that the Fowler Brothers (who authored The King's English) gave to structures such as the ones italicized in the following sentences:
- Women having the vote reduces men's political power. (Fowler's example)
- He insisted on them coming.
- I read about him winning the race.
We can talk about that structure if you like, but it has nothing to do with the question with which you began this thread.