I haven't been able to find anthing about a private theatre at Strawberry Hill, but I did find this interesting tidbit about Walpole and stage craft:

Robert Jephson's The Count of Narbonne, the first Gothic drama, was a dramatization of The Castle of Otranto.

Gothic drama was especially 'spectacular', and achieved some notable successes. The setting of Act V, Scene VI of . . . The Count of Narbonne (first performed at Covent Garden on 17 November 1781) was not untypical: 'The inside of a Convent, with ailes (sic) and Gothic arches, part of an altar appearing on one side; the statue of Alphonso in armour in the center. Other statues and monuments also appearing. Adelaide veiled, rising from her knees before the status of Alphonso,' Walpole himself supervised the production, instruction the actors, adjusting their costumes, even loaning medieval garb from his own collection. Walpole's arguments with Jephson over the set--Walpole felt it was more authentic for the status of Alphonso to be recumbent on his tomv, whereas Jephson wanted it should stand erect--led to their estrangement. *

I know that he wrote at least two plays: Nature Will Prevail (a comedy) and The Mysterious Mother, a Gothic drama.