You automatically own the cottage and any other improvements added. There would be no further title or deeds.
Did you recieve a deed and title insurance at the time you bought the property for cash? If not RUN, do not walk to your attorney.
Or, provided you have the deed and title insurance from the purchase of the land, you own the cottage too.
Additional concerns are:
Did you or the builder obtain permits? Was the structure inspected and approved upon completion by the local building inspector? A simple phone call to the city inspector should confirm whether a certificate of occupance is in the works.
If so, you probably have nothing to worry about. You should get a receipt from the contractor and his subs showing the work was paid for in full. Otherwise, you could possibly have a mechanics lien placed on your property in the future by some unpaid subcontractor or supplier.
Also, for these concerns, please seek legal advice from a licensed attorney in your state. A home improvement message board is not a good source for difinitive answers in these matters.