#1's idea is very valid :)
Utilising Beziers as a starting point: another random itinerary:-
1) Grab a train to Avignon (2 hrs direct train), see that for the rest of the day then most of the following morning then in the afternoon hire a car for another 2.5 days to explore the local Provence area (for example Saint-Remy-de-Provence, Les Baux and Moustiers Sainte-Marie. Dump the car in Nice once done.
4 days done, 12 to go:-
2) Grab a train from Nice to La Spezia in Italy (5.5-6 hours) to spend 2 days in Cinque Terre strolling from village to village and getting the train back (Cinque Terre is small. Warning: it'll be busy, but that's cos it's very pretty).
3) Grab another train from La Spezia via Milan to Como (4 to 4 1/4 hrs) and spend 2.5 pleasant days in the Lake Como area.
9 days done, 7 to go:-
4) From there continue North into the Alps. Again, train time: (4.5-6 hours), various connections will take you to Interlaken in central Switzerland but I would go for one of the villages in the mountains around it - pick from Lauterbrunnen, Murren, Wengen, Grindelwald. Again, 2.5 days here. Plenty of walking or if you're not so fit there are various cog trains, lifts and funiculars to get up to the peaks.
12 days left, 4 to go:-
5) Back to Interlaken; more trains:- Grab an IC or ICE to Mannheim, connect for a local S-Bahn to Heidelberg (4 3/4 hours), see that for 1.5 days.
2 days left:-
6) Grab a train to Mainz and spend 1.5 days leisurely exploring the Rhine between Mainz and Koblenz to the North. This stretch of the Rhine is UNESCO World heritage and rightly so. Use https://www.k-d.com/en/home/ for a boat trip with running commentary. Trains to hop stretches are frequent on both sides. Consider going up the http://sesselbahn-boppard.de/ in Boppard for an amzing view. Once done, take yet another train to Frankfurt airport to fly up to Copenhagen
Links not provided already above:
French trains = http://www.voyages-sncf.com.
Italian trains = http://www.trenitalia.it.
Swiss trains = http://www.sbb.ch.
German trains = http://www.bahn.co.uk.
Timetables can be sourced from the German railway site. If crossing a border, check the prices for both country A and country B as they can vary. Definitely book in advance wherever you end up going for the cheapest price (preferably now). For additional information about what to expect on trains, refer to http://www.seat61.com. For a hire car company in France, the major players are Avis, Hertz, Europcar, Enterprise and Sixt but I was too lazy to check which ones are based in Avignon (Google or their own websites can easily tell you this). Bear in mind that all of these places are in high season so consider booking accommodation in advance.
If you can cheaply change your flight home out of Frankfurt instead of Copenhagen, would make things much easier for you with my itinerary (or consider using Madrid or Barcelona's airport for #1's intinerary if you prefer heading down through Spain). You'll have to spend a fair bit of € to get back up there no matter what, flights can be found via http://www.skyscanner.net