And welcome to the wonderful world of touring on a bike!
I'm going to go back to basics....
You want to
1)Cycle to Trieste for April 25 (1900 km)
2) Run 10k on April 27
3)Decamp to Geneva for May 12 (By bike? 900 km)
4) Run a marathon May 12
No information on how you get back...... are you expecting to survive?
Can you do it? Yes. Will you enjoy it? That's up to you.
I'm going to assume your fitness is good but I wonder what a 1900 km tour would do to you physically that would negatively affect your running goals? How would you feel if your cycling meant that you couldn't complete your marathon?
As suggested above, talk to some with experience.
I would assume that a a first tour of almost 2000km across several countries (how are your languages?) on a deadline on a new to you bike could be stressful. One day of heavy rain you could sit out. But 4 or 5 in a row could seriously dent your progress. Mechanical problems could put you under pressure too.
As suggested above, you can always consider a train to catch up.
Budget: Have you thought how much this could cost? I'm assuming you're not camping so you'll have hefty nightly costs unless you use something like couchsurfing or warmshowers (couchsurfing for bike tourists). But these will take time to organise and I think time is something you are very short of. In any case, work out a daily cost - food, accommodation, misc to get an idea.
What a bloody great adventure!
Now some suggestions
As said above, get out on a bike... now. Any bike. Get a feel for what you can do. Aim to work up to a couple of shakedown rides that will tell you what distance you can do, what clothing works and what you need to bring with you.
Think about what you want to bring and how much space that will need. If you really want to, staying in hotels/B&Bs you could get away with a small bag, washing your clothes every night. More comfort means more gear.
Now think about the bike. Is this a one off or will you want to do more? If you plan to do more then it is worth it to choose carefully. In any case, I think the 2 most important things for you are that it has to be comfortable and it has to be mechanically sound.
Excellent advice above to pick up an old Mountainbike and have it converted. (Rack, mudguards, dynamo etc) Cheap & solid. (I have done that).
Again, budget is important. And don't forget things like panniers to carry your gear and some decent clothes to cycle in. I don't necessarily mean cycling clothing.
If you don't know anyone who is good with bikes, check out your local shops (when they are not busy) and seek advice. The problem with some bike shops is that they don't listen. If they don't listen... walk.
Worst case, you'll be travelling in a civilised area and lots of help will be available if you do have any technical problems.
One of the great advantages of travelling by bike is that people become very helpful!
Think about navigation. How will you do that? Maps? GPS? Phone or dedicated device?
Great that you're looking to get experience in Decathlon.... but be warned that some mechanics in these places are good - others not so.
And you don't really need off-road tyres. Most of your route will be on good surfaces.
Now, I'm going to turn things on their head... literally.
How about doing your tour.... in reverse?
Get you bike and gear and fly there? Then load up and cycle home.
I've two reasons for my suggestion.
The first is that the pressure is off.
The second is that you will be cycling through some prime touring real estate and it seems like such a shame to rush through it. People travel from all over to cycle along the Rhine or to cycle in the Alps. You'll be rushing through.....
If you fancy camping you could save a bit of money.
And snow will have less of an impact!
In any case, the very best of luck with your investigations!