Lauren,

Most of the info came from Era Magazine, supplemented by other newspaper articles.

Mary Ann and Dot are definitely not the same person. Here's a chronology relating to the Gaiety Theatre of Varieties, Hanley:

Feb 16 1889 - Mar 16 1889
Proprietress, Mrs Geoghegan
Directress, Miss Dot Geoghegan

Apr 13 1889
Proprietress, Mrs J.B. Geoghegan
Directress, Miss Amy Geoghegan

Apr 27 1889
Proprietress, Mrs J.B. Geoghegan
Directress, Miss Amy Geoghegan
"Closed for extensive alterations"

Jun 4 1889
Mary Ann Geoghegan dies in Hanley.

Aug 10 1889
Lessee, Mr Chas. Hearn
Acting Manager, Mr Barry Ivor

There is no further mention of Amy/Dot Geoghegan (who is probably one and the same person). It's possible that Charles Hearn was renting the place from the Geoghegan family.