Can’t speak for Bratislava but I like your other choices, especially Strasbourg.
About 30 miles away from Strasbourg is France’s only WWII death camp, Natzweiler-Struthof.
Do you need or want to have a car the entire time? I’m just not sure it’s necessary.

I like your choices of cities and stops and 35 euros should cover food, admission fees and the occasional beer and wine.

I wouldn’t drive for this itinerary. I think all of your along the way stops are easy day trips from the cities you plan to visit?? For example, Dachau is a Munich suburb — easiest to get to using U-bahn and bus. Neuschwanstein castle is near Munich, doable as a Romantic Road day trip and maybe a place you will want a car. Have you looked into the drive times involved?

Crossing the English channel:

Best, without car — Eurostar, the chunnel train. It can be relatively cheap. Look for low fare specials at www.eurostar.com or any other travel agency that sells European rail tickets.
Next best, without car — look for a cheap flight. Use www.whichbudget.com to see which low fare airlines fly which routes and check our “cheap flights” forum for more tips.
Last choice, without car — train or bus/ferry/train or bus. Usually not enough cheaper that the other options (assuming you book well in advance) to be worth it, IMO, since it kills a whole day.

Best, with car — Eurostar, but I think this is terribly expensive to take a car. If you’re renting a car, I think there are some rental options that combine a car in UK with Eurostar as a passenger and another car in France.
Next best, with car — ferry. Probably from Calais to wherever in England and from Dover to Ostende on the way to Brussels — you might consider staying in Brugges instead of Brussels.

Well, thanks for all the info. We´we been thinking about skipping London, probably going next year only there. Maybe adding Budapest and Belgrade instead and reducing the costs in the process for about 100€ per person.

So, if my math is right, 35€x20=700€ for food,drinks,admision fees; 100€ for fuel and toll costs; 15€x15=225€ for hostels. So with about 1000-1100€ we should have enough per person probably?

Also, anyone had experience where the best and cheapest way to leave your car in big cities like Paris, Berlin, Amsterdam,…

Well, thanks for all the info. We´we been thinking about skipping London, probably going next year only there. Maybe adding Budapest and Belgrade instead and reducing the costs in the process for about 100€ per person. Also, anyone had experience where the best and cheapest way to leave your car in big cities like Paris, Berlin, Amsterdam,…

I’ve never had a car in any of those cities, we’ve only ever had a car in France when we were leaving Paris.

In Berlin I think you can find free street parking in the suburbs. In Amsterdam you should park it outside the center, in a Park& Ride facility for example.

For Paris, I am not sure. I guess find a cheap parking garage.