It is easy enough to drive and park anywhere in Europe (except maybe Gibraltar and Vatican City )
There are a few issues you need to build into your itinerary though.

The roads in some of those cities are more congested than you may have encountered before.
Driving from one side of Paris or Brussels to the other is at least as bad as driving from one side of LA or New York to the other via the busiest points.

Your itinerary takes you across the alps at least once. Most passes through the alps (even by freeway) have windy 10% or more grads where traffic slows to a crawl when the freeway is only two lanes wide and trucks block your path. Plenty of snow in the alps in April. A 100 mile drive could take 5 hours. Ever driven in and out of Tahoe in the snow?

European 'freeways' tend to be built through the open country and don't pass that close to the towns. Allow plenty of time for that. Main roads don't go through the little towns either, you have to leave the road to get into them. You can drive for 500 miles without passing through a town at all, no problem Going from Brussels, drive via Cologne and then alongside the Rhine all the way to Wiesbaden rather than the freeway. Will take a few days but that way you will see all the little places. That is a drive I would like to do again.

The places that are not frozen will still be cold and very wet (April is usually the wettest month in Europe)

Zurich (Switzerland) is not in the EU. Check that your rental car is even allowed entry into Switzerland and that you have a visa. Driving in and out can be like the US to Mexico.

Two weeks is doable but you might not get as much spare time as you think. Can you say what your itinerary is so far?