The answer to your first question is yes. ..if you clearly have it indicated on the search order. So long as there is some clearly stated method of calculating the date of payment for the court. Either a particular date or an enumeration of the days from the date of the search will work. However, make certain that it is stated on the search order since it is authored by the client, and is then admissible into evidence as an admission by a party opponent.

The answer to your second question is no.  It could be argued by the client that you are altering the terms of the agreement. Although... if they accept your abstract , and use it for the closing/title insurance it would give rise to an argument that they have accepted the payment date you included. Also if they accept the benefits of the contract they are equitably estopped from denying the validity of the contract.  Stating the date of payment on your abstract would not completely make your invoice noncollectable, but it would make it more difficult than having the term of payment stated on the search order issued by the client. The signing agents do it routinely with respect to the confirmation (job order) when they are contacted for a closing.

Remember also that the abstract is your property until you are paid for it. They get very upset if you tell them to return your property unused or pay you by a certain date, but if they are a slow pay/no pay client...what difference does it make?