Switzerland in March

Yes, absolutely!
You can also ski in the Berner Oberland, by the way - March is still great for that. The terrain is lower and mellower than in Zermatt, but still very enjoyable. And a ski pass will be the cheapest way to explore the various viewpoints (Jungfrau not included)!

How early is "early", for your flight home? If it is later than 10 AM, I would suggest that you stay in Lucerne at the end of your trip, and go to the airport directly from Lucerne. That way, you "save" a night, which you could spend in Zermatt. It would be good to have 2 full days there, because if it is windy, the skiing area gets significantly reduced (you quickly lose the entire Schwarzsee / Klein Matterhorn side), and if it is very cloudy, the area is largely treeless so visibility is poor.

are other areas open for visiting/hiking?

You can find winter hiking trails at Switzerlandmobility

What kind of visiting are you thinking of? Castles and museums, or something else?

Not too many museums and castles. Wandering around, talking to people, having a drink at a pub, outdoorsy things.

Depending on how firm your dates are, a very fun event in Lucern is Fasnacht (carnival) on the 16th - 21st of Feb. Lots of music, people outdoors wandering around, lots of opportunity to visit with others during this shared experience. I went to it several years ago just for the day and loved it.

Zurich also has one Feb 24-26.

I can only go during spring break, as I'm a teacher. March 10-20, roughly

Bigsix, what itinerary did you decide om? My son and DIL just decided they’d like to do Switzerland on their spring break, week of March 6. I’ll be doingvthe planning. My husband and I stayed inMuerren a few years agoo, but in early June.