Well, there's an incredible lot to do in all of the countries you mentioned, and the general route you've suggested is a great one - no backtracking, which takes up valuable time.
To your general questions: 1/ Yes, it's possible. And it's a good plan.
2/ It could take anywhere from two weeks (highly not recommended) to the rest of your life, depending on how long you have. A month or 6 weeks would be a great, relaxed time frame to really check stuff out.
3/ How to get around depends a lot on your budget and time. If you have lots of money and little time, fly between hubs, and take trains within countries. If you have more time, train is a great way to get around between countries, too. If you have the rest of your life, riding a bike or walking is cheap and you see more.
It's harder to answer your last question, because none of us know what sorts of things you and your friend like to do, or want to see. There is an overwhelming amount of information out there about traveling, and it can be pretty daunting to try to figure out what you really should do and see, but the time you put into planning really does pay off when you're on the road, seeing what most interests you. For instance, I could put together an incredible itinerary for you, focusing on Mozart's life and times, but if you don't care a fig about Mozart, you probably wouldn't enjoy the trip a lot.
A good way to start is to look at the things that interest you at home - do you like going down the pub and chatting with friends, or would you rather go to a football game? Maybe hiking is your idea of a perfect weekend, or you'd prefer to go to a museum or art gallery. Maybe you can't stand the National Gallery, but you'd drive 100 miles to see a photography exhibition. Whatever it is that you like at home, chances are, you'd like it in another country, too. Think about what you liked at school, too - history class? music? art? literature? And are there any iconic buildings that you would just love to see? Maybe the Eiffel Tower doesn't appeal, but you loved the movie Gladiator. So put Rome on your list, and check out the Colosseum.
Once you've got a general idea of what interests you, check out guide books and go online and try to find things that fit in with those interests. It does take some reading beforehand, but that research really pays off in the long run. And then, when you have a little bit of a plan, come back and start peppering tpunks with questions (read some of the topics that cover the places you're going, to get all sorts of tips and treasures, too).
All of the countries that you mention have rich history, art treasures, amazing musical traditions, stunning architecture, magnificent landscapes and outdoor pursuits, appetizing food and drink, and friendly people, so you really can't go wrong - it is just a question of figuring out which of these you want to explore.