1.) Is a person able to change their address at the polling location?

Yes.

2.) Is a person able to vote at the new address at that time?

  No.

3.) Why are they are not able to vote at the new address at that time? 

As per Texas Election Code Section 15.025. the registration of the voter whose information is changed on the registration records becomes effective as to the change on the 30th day after:

(1) the data the voter submits to the registrar a notice of a change of registration information under Section 15.021 or a response under Section 15.053, indicating the change; or

(2) the date the voter submits a statement of residence to an election officer under                  Section 63.0011 or a registration application or change of address to an agency
employee under Chapter 20, indicating the change. 

Summary:
After the voter submits the change of address, or Statement of Residence (SOR) at the polling location, the Elections Department processes the change and sends the information to the Secretary of State's Office (SOS) so that they can update the state database.  On or before the 30th day of making the address change, the Elections Department mails the voter a new and updated Voter Certificate and now the voter will be able to vote using their new address.