If you buy a $100,000 house and finance the purchase for 30 years at a rate of 3.75%, let's see how PMI will effect your payment:
Down Payment 20%: Loan Payment: $370.49, PMI: $0 = Total Payment: $370.49
Down Payment 10%: Loan Payment: $416.80, PMI: $37.50 = Total Payment: $454.30 Diff: $83.81
Down Payment: 5%: Loan Payment: $439.96, PMI: $39.58 = Total Payment: $479.54 Diff: $109.05
Down Payment: 0%: Loan Payment: $463.12, PMI: $41.67 = Total Payment: $504.78 Diff: $134.29
As you can see, when you factor in the PMI, the monthly payment increases dramatically.
Your research regarding down payment is incorrect. The amount by which the monthly payment is reduced depends on: (1) the interest rate; (2) the amount you're putting down; (3) the length of the loan, and other factors. There are plenty of online calculators that can show you how those variable work together.
As for those up-front costs, many (but not all) sometimes can be included in the loan. It depends on the particular type of mortgage. A buyer also can ask the seller to pay some of the closing costs. That's a point of negotiation; your Realtor can help you with that. But, to oversimplify a bit, it's often possible to purchase a home and pay little or nothing in closing costs.
Hope that helps.
There are programs; fewer than we used to have, and sporatic at best.
Also, understand that as a rule, if you have less than 20% equity, the Lenders require you to have PMI, which insures the Lender in the event you default: This can add a lot to your monthly mortgage.
If you put 10% down, then you will pay PMI until you reach that 20% level, when you can petition the Lender to stop doing that.
Talk to several Lenders; look at their GFE's and compare the numbers.
Good luck and may God bless