credit checks
Yes if it for a mortgage application. It will drop an 800 FICO by 5 points and a 500 FICO by 1 point. There are consumer protection laws that allow for shopping within a specific industry for 14 days. However, the formulas that calculate credit scores rely on credit providers to prescreen and delete...

If debt ratio is high and no equity in home, but credit score and employment is good, is refinancing impossible?
Give me a call! I have been doing this for 30 years...

Should I get a piggyback loan to avoid PMI or will paying PMI with lower rates be cheaper?
You can get Lender Paid MI and loose the monthly MI payment. With rates this low it may increase your rate a little.