IMO, Appenzell is worth a visit as it's a somewhat different part of the country with a unique culture. As far as hiking, if I were in the same situation I'd pack along some good rain gear and just dress according to the weather. When I was last there (during June a few years ago?) there was the possibility of snow at the higher elevations so I didn't bother going to Ebenalp. However that didn't matter as there was some kind of a local event taking place at the time, to commemorate the burning of the city 500 years ago, and there were people wandering about in suits of armour carrying Halbards, Swords and other gadgets. There were food stands setup, music, etc. and it was all quite entertaining. I also attended a local folk music concert one evening at City Hall - https://www.youtube.com/watch?v=HdNZ8Pa03ko . You may find this interesting - https://www.appenzell.ch/fileadmin/template_appenzell/user_upload/06_Dokumente/Broschueren_Flyer/Appenzeller_Braeuche_und_Traditionen_EN.pdf .
I would recommend at least two nights in Appenzell (preferably three) as one night doesn't allow much time to look around.
Lugano is a somewhat different experience as it's a somewhat posh city right on a lake. While there you could visit the Casino, take the Funiculars to the top of local mountains or take a boat tour on the lake.