The president at his own discretion submits the budget to the House through the OMB. The president at his own discretion signs the budget after congress sign it.
The Congress can't rewrite what is submitted. The congress can refuse to sign the appropriations.
Reagan submitted and signed those federal budget
Yes, they can. In fact, they can totally ignore anything the President submits if they want.
I am not sure you and I should debate this matter any further.