In May, I would think of the swiss alps - wonderful walking, using the trains to get about, beautiful alpine flowers and scenery. alternatively, Crete, or Northern spain, where the flowers are also lovely at that time of year.
Or here in Cornwall [south-west england] for the same reasons - it's often the best time of year and it certainly isn't crowded.
congrats & good luck!

OK. May.

Hmmmm.

"She hates museums and swimming and I like to hike and enjoy other cultures."

I say Switzerland. Depending on the length of the honeymoon, you could visit more than one area of Switzerland (or Switzerland plus another country).

Switzerland has loads of "places with a lot of good walking/hiking tours involving beautiful landscapes, churches, castles" and in May, if won't be super crowded and touristy if you choose the right towns to stay in.

If you enjoy European cities, I recommend Copenhagen. We just returned from a week in Denmark, and a week in the Scottish Highlands. You can walk in Copenhagen, or catch a bus or train to Sweden. There is shopping, beaches, and Castles to see. In Scotland, we hiked the Birks of Aberfeldy. We drove our small car to many villages, then parked and walked the streets. We enjoyed a cruise on Loch Ness. There are lovely B&B's in every village.