Yes, the executive branch does submit legislation to the Congress. In fact, the current administration missed a required submission deadline for the budget which is just a big bundle of legislation.
Please learn how the US government works before the next election.
Yep, that is how many programs get funded. It is a line item in the budget. If it is not a line item, then it doesn't get funded. Programs are funded in the budget, then an RFP goes out, then final regulations, then implementation. So yes, without the line item, no program.