If I activate an SOS using my Inreach Mini does this appear on Mapshare? If so, what does it show?

I have no idea. If it shows, I would imagine that the messages are what would show up.

If you are concerned that the folks at home will not know you have triggered an SOS, the best bet is to send a specific message to MapShare with that info. Also note that GEOS will contact your "emergency contact" when they receive the SOS.