Seller has requested us to extend settlement date by a month. Is this a common occurance? What costs would be involved in such a request and what would the seller be required to pay?

There could be a variety of reasons why the seller would request an extension.  The costs you might incur would be the interest rate on your loan if you have locked and it were to expire.  Check with your lender and if this is the case ask the seller to pay your cost if you agree to the extension.

One alternative might be to close on the purchase and lease the property for 30 days to the seller.

Kent described what was/is in our Agreement of Sale. Before signing our AOS for the sale of our townhouse, I had our realtor put in it that if we could not find a new house then we could rent back our townhouse at an agreed-upon price.

The theory was that we were concerned we might not be able to find a house to buy quickly enough. Turned out we found one so we didn't have to use the 'rent-back' option. The buyer of our townhouse wanted to move in as soon as possible, so she would have been pretty upset had we had to stay longer.