Did Arcane's house in Lahaina also burn down in the fire there?

Possibly. However, remember he still has a place on Mt. Shasta and a place in Switzerland as well as many places through time on both timeline 1 and timeline 2 from about 1500 AD to 7000 plus AD on the first timeline. But remember timeline 1 and timeline 2 were the same timeline from before the time of Christ until September 11th 2001.