I see you have received some good advice about your upcoming trip. I have been traveling to Europe for over 35 years as a travel writer and I have imported various pieces of art. You asked lots of questions and I will add a few of my suggestions....4 to 6 weeks in Europe is really a very short amount of time to see many of the places you mentioned. You need to ask yourself if you really want to see as many different countries as you can in that span of time....or do you really want to feel the culture of the countries you visit. I always suggest to pick just a couple of countries and and visit as many parts as you can. One country Italy, has much more than you could ever see in 6 weeks. The hassle of traveling and learning the secrets of each country is demanding in itself.
You could use the rail system to your advantage with a pass. I always use the website www.raileurope.com to plan my rail trips. Many of my trips to Europe are 4 to 8 weeks and I spend most of my time in one area and see all that it has to offer. For example, my last trip to Cinque Terre, Italy....I stayed for 5 weeks in Monterrosso and made day trips and went to 60 % of the areas I wanted to visit and always returned to the same hotel. I have much more I could tell you, but you need to decide which countries you want to visit. I would suggest Italy and France and you can never see it all. When you decide which countries to visit, you could invest in Rick Steve's travel guides, which I think are some of the best on the market to tell you how to see it all and what is a real waste of time. There are lots of cheap hotels, hostels and rooms for rent to keep the costs down. You can also travel on the train at night and pick up time and save money.
But, if you want to see all those places in just a short period of time ( 4-6 weeks) I am not the guy to be suggesting anything. You need to make time and find a way to go to Europe each year and do something different. You will have better experiences when you travel this way...other wise, it will be just like traveling on a fast pace tour that tries to see 10 countries in 3 weeks.
Good Luck, Tom