a) The ability to refinance means that a fixed-rate loan with no prepayment penalty is really a fixed-income security with an embedded option. Is this a "callable bond" or a "putable bond"? In practice, is it American-style, European-style, or something else? Who buys the embedded refinancing option and who writes it? What is the strike? 

How would you value a bond with an embedded option, compared to a bond without optionality?

A borrower who finances a purchase of a home, car, etc. with a fixed-rate loan also has an "embedded" option to refinance their outstanding debt when market interest rates decrease. (Most mortgages, auto loans, etc. do not have "prepayment" penalties.) A borrower buys a house or car using some combination of cash (for the down payment) and borrowed funds equal to XO. Sometime after the origination of the loan, the principal outstanding is Xt < XO. If interest rates have decreased substantially, the borrower may choose to borrow Xt at the new, lower market rate and pay off the original loan, effectively replacing a high-interest loan with a lower-interest loan on the same principal. This process typically requires that the borrower pay some "closing costs" on the new loan (legal fees, appraisal costs, or some amount of "cash to close"). The borrower will choose to exercise the refinancing option if the present value of the savings on the remaining interest payments they will owe on the loan is greater than the closing costs for the new loan.