I'm thinking about applying to several mortgage brokers and lenders for a refinance, then choosing the best mortgage rate quote and locking my loan with that lender. Is that allowed?
You can apply with as many refinance lenders as you like. Just understand it's not like applying for a car loan or credit card, and each mortgage application can be time consuming and a bit costly.
Be considerate of the loan officers, processors, underwriters and title company staff, and try to waste as little of their time as possible. Be efficient by making copies of all of your income and asset documentation and including it with your applications. It can really drag things out if lenders have to keep coming back to you for paperwork. Ask each lender for a credit approval or mortgage pre-approval, and explain that you don't want to pay for an appraisal until you know that you'll be approved to refinance.
Credit reports may cost you about $15 each. If all lenders check your credit within a 14 day period, the inquiries are treated as a single inquiry for FICO scoring purposes.
Once the lender(s) approve your application, you'll probably want to get another set of mortgage quotes and choose the mortgage lender you'd like to complete your refinance. Cancel your applications with the other lenders at this point. Your chosen lender will need a check from you for your appraisal. At this point, you might also want to lock your refinance rate.
FHA mortgages are a bit different. Lenders are required to apply for a case number with HUD when processing your application. Make sure that no one orders a case number until you have decided which lender you'll be using. Case numbers can be canceled and re-issued, or transferred between lenders. However, not all lenders are as efficient or cooperative about doing this.
What about getting several appraisals?
If you worry that your property appraisal could come in too low for you to refinance, you might want to pay for several with different lenders. However, you can probably accomplish the same thing by going forward with only one lender. If the appraisal comes in too low, you can always reinstate your application with one of the "runners up" and try again. Remember that every application starts a process in which teams of employees work to get your refinance processed, approved and closed. You can be considerate as well as careful with a little effort.