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For monthly cost, Conventional program is better in your situation. For FHA even you put 20% down, it is required to pay .85% monthly as Mortgage Insurance and also 1% up front mortgage insurance if you go with 30 years fixed program. However for FHA program, you could receive up to 6% seller's credit compare to 3% max for conventional program.
So the bottom line is unless you are purchasing a condo, go with conventional loan. Also for the last question of yours yes you would have to choose the program first and lock the rate later.
If the %20 percent down payment is available and your desire is to establish the lowest monthly payment, then go with a conventional loan.
As far as the lock goes,. some banks have a fee to lock a rate. Other potentially losing an application fee, a buyer can change m mortgage company if they find a better deal and the current broker won't compete with.
Depending on the lender you may be able to change programs after a rate lock but it may delay the closing.