Q: The lender is asking me why I am buying a smaller house, if it is closer to my work and why I want to buy this home. The lender is even requiring a letter with the reasons before they will approve the loan. Can they do this?
A: While you don't provide nearly enough information, the short answer is "yes." For example, you don't mention if this is a primary residence or a second home; there can be different lending rules and mortgage rates for different kinds of properties. Are you retaining your existing home and buying another?
If so, both can't possibly be your primary residence, and so the rules which govern your purchase would likely apply to second home (or even investment property) mortgages.
Lenders are rightfully concerned about fraud, and certain kinds of fraud (such as occupancy fraud) are far more prevalent than you might think. That's why your lender might ask you questions about your intentions, especially if they are outside the norm, such as moving farther away from work, or buying a smaller home when you already have a larger one. Trying to assess your intentions allows them to apply the proper set of rules for your transaction, helping to protect them in the event that you should fail to make payments on your loan.