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To find out timetables, I suggest you visit either www.bahn.de
(German railways) or SBB: Der grösste Bahnhof der Schweiz.
(Swiss railways). Both of them (or at least the German site) will allow you to only select trains that will accept your bicycle.
The simplest and fastest options include TGV from Paris then ICE in Germany. Only some TGV will carry bikes, and ICE will not. I am not sure about exactly what is accepted when you bike is not mounted.
TGV require advance reservation for both you and your bike. For TGV bookings with bike I am made good experiences with www.trainline.fr
TER in France will take bicycles for free with some restrictions (peak hours in certain areas, and ofter the space is tight).
In Switzerland, regular trains will carry bikes for a charge. ICN trains will require a bike reservation in addition to this (but no reservation needed for yourself)
In Germany, regular / regional trains carry bikes for a charge, probably without reservation. I expect Austria to work along the same principles.
So it is hard to review all the possibilities, but to summarize : either you pack your bike in a way that is accepted in all trains (this is easy for a bare road bike, slightly less for a tourer), and you travel fast with minimum connections. Example is Paris to Linz in 10 hours with one change in Frankfurt. Or you want to carry your bike as is, and you jump from regional train to regional train. It is probably still possible to make the trip in one (long) day, though.