Very interesting set of questions! Some playbills record benefits for local causes like dispensaries, but not usually for major disasters like this. A very cursory search of BNA for the Scottish crisis in 1837 has returned reports of a lot of fundraising initiatives in Britain and Ireland including the Channel Islands, mainly from local civic corporations and organisations, church collections, royalty and the occasional aristo, but not much in the way of the theatre. A proposed fundraising ball at the King's Theatre in London for 28/07/37, organised by various dukes, earls etc https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18370603/020/0003
was put back a few weeks and moved to a different venue, and the hoped-for attendance of various royals doesn't seem to have taken place. More relevant perhaps is the donation of 10 guineas by Charles Kean in April 1837 while at the Theate Royal in Edinburgh, with the theatre manager donating the same sum. Perhaps other Scottish theatres raised money?
How influential was Kean at this time? As you say, something worthy of research.