is it compulsary to have a dual entry i between 1st solo and 2nd solo?

Answer 1

Once your instructor has signed you off as competent to solo in the aircraft, you may solo again, before any more instruction is given.
The instructor, or owner of the aircraft, may require some more, but it not required by the FARs.
The only reason any more endorsement would be necessary is if you didn't go flying for 90 days, then you would have to have your book endorsed again.
When I was a student pilot, I didn't get to fly much because of my job and 2 times I had to have my log book endorsed to solo because I didn't fly for 90 days.

Answer 2

no

Answer 3

No.