Do I Need To Submit Form 16 If I Am Joining New Company In 1st April

Currently my company does not deduct TDS, as it is a small company hence has no Form 16. Also I didnt have to pay tax as after 80c deductions I was under the tax slab. And have also not filed returns.
I am about to join a new big company. Now if I join in 1st April will they ask for Form 16 or any other tax related documents.

Normally no company will ask for your previous Form 16. But if they have any doubt in your take home/gross salary, for having further clarification from your Form 16, they may ask for the same. You can even pay income tax on voluntary basis.
If you pay your Tax, then with that you can even avail Housing Loan.