In NY the title company will examine the title.  Title is ordered by the bank or broker on a refi and by the attorney on a purchase.  Buyers rarely order title. A survey affidavit is usually used on a refi.  On a purchase it would depend on the purchaser's attorney preference and how old the survey is. 

Price of a survey, would depend on the county.  A good average would be $600 - 850.

Each area has different customs as to who does the closing.  The majority of NY closings is done by an attorney.  Most other states are closed by the title company (or an independent closer for the title co).

Hope this helps.