does down payment affects the mortgage overhead cost?

Not sure what you mean by "overhead costs" but the less you put down, the higher the interest rate. In addition, if you put less than 20% down you will have PMI factored into your monthly payments. Both of these will impact your monthly payment after closing.

At closing, your pre-paid interest will also be higher because your loan balance and interest rate will be higher.

If you make less then 20% down the for your your monthy payment will be higher due to PMI.

Really that wll be your overhead once you close your loan in addition to your principal and interest payment.