On #1. No it does not work that way. Since you are on the fixed rate, your payments right now are spread over 30 years, they never change, so your payments do not get lower because you have a lower principal balance. Even though your principal balance right now is spread over 30 years, the difference between 28 and 30 will be completely negligible and is not the reason why your payments are lower.
If your rate is 3.75% maybe you should look for a lender that will do 3.5%. If your credit score is over 720, your loan amount is over $150K and you have are refinancing a single family home, you should aim for 3.5%. If you can get a lender to give you that, that should be the determining factor which way you go.
Why does everyone want you to do this? Well they will make money from servicing this loan for 30 years. They will make a little money upfront from securitizing this loan but the big money is from you making the payments. It is a relatively easy loan for the lenders it takes little effort, it is a no risk loan since the loan is guaranteed from the HUD (if you do not pay they will cover the loss), and there are not so many guidelines a lender needs to worry about.
QL is the Walmart of Mortgages. They are quick, have generally poor customer service but offer decent rates. They are generally pushy but even more pushy with streamlines as these are the best deals to get right now, and they know they could lose you any minute. One thing to remember is that FHA loans have an FHA case number assigned to the property. So if you have started the process with QL they have pulled a unique case number to your property and no other lender can pull another one. This is the only number assigned to your property for 6 months. Now if you want to use another lender because you get a better deal, you will have to get this case number transferred to the new lender. QL might try to stall the transfer to get you to stay with them, but if you do chose another lender do not get discouraged, if the deal elsewhere is better keep pushing for the transfer, they have to do it. 3.75% however is a decent deal.