Can I ask a theater to allow me to bring my own interpreter and request designated seats so she can sit in front of me? I would also like a copy of the script ahead of time so my interpreter can read it.

The answer here is “yes, you can bring your own sign language interpreter.” You can bring someone with you as your companion or interpreter but you would have to buy two tickets.

BUT the requirement for effective communication does not require the theater to take any action that would cause a fundamental alteration in the goods or services being offered.  Having someone sit in the seat in front of you so she could interpret would disturb the other patrons and would fundamentally alter the experience for them. 

However, asking the theater to hire a sign language interpreter does fall under the effective communication requirement. 

If the theater provides the interpreter, you would only pay for your own seat.  The theater would cover the costs of the interpreter and would also find a seat for you where you could see the interpreter clearly.

Giving a copy of the script to non-theater employees may violate copyright and other contractual provisions agreed to when the theater obtained permission to put on the play.  So that request would probably have to be denied for someone who was not employed by the theater. A colleague of mine provides audio-description of plays for patrons who are blind.  She is a contract theater employee, so she is allowed to have a copy of the script while the play is still in rehearsal.