Here's my humble opinion:
" Here is what I have found while searching the net".
1. How many sentences are there in the above sentence?
There are 2 clauses.
2. What is the name of a sentence formed of two or more sentences?[ a compound sentence?]
I think yes.
3. What is the grammatical function of "searching" in my sentence?
An adverbial modifier of time.
4.How can I rephrase the last part of my sentence? Could it be:
a] ",,while I was searching the net"
b] "while I searched the net"
I think either is possible.
and what is the difference between [a] and [b] or which of [a] and[b] is more correct?
To my mind, there's very little, if any, difference. I'd personally prefer a].