You've posed quite a few questions and it is clear you have done very little (if any) research on the subject.
My advice would be to browse these forums, check out
www.crazyguyonabike.com, and come back.
I've never toured Europe by bike but I will attempt to get you started.
1) 600KM over fairly flat terrain (your assumption, not mine) would take anywhere from 6-10 days depending on what kind of training you have done prior to the trip. If you're really out of shape - could take longer than 10 days. (You might want it to take longer than 10 days...I would)
2) VERY rough estimate: 750$ to 2000$
3) Can't answer.
4) Plenty of people in France speak english, though you shouldn't assume they do. Learn some basic phrases in the native tongue (hello, good morning, do you speak english) and you will have an easier time. This is good advice for anybody outside their native land.
5) If you have time and resources - why not?
6) Don't know - I think it's all about location. Use your best judgement.
7) Bring only what you NEED. Figure out what you need by doing some overnight trips before you head to europe.
8) Use maps.
9) Look at the used bicycle market. A NEW touring specific bicycle is likely to cost a lot more than 300$.
10) You could bring the kitchen sink - but do you really want to lug it all around europe?