It is a good idea. Not necessary though in order to access health care. If you are in an environment with lots of expats or generally mixed backgrounds you'll likely survive without speaking much more than a handful of German words per week if you really want to (Heidelberg, Berlin, Bonn, Cologne, ...). It does help you though, as finding doctors and especially employees in public office who are able and willing to discuss with you in English is not a given.
It's possible, but it severely limits you if you don't speak German. I've been working with large companies - either international branches or German multinationals - and they don't like to speak English at work. They can, but it is dumb to have ten German engineers to discuss a topic in slightly broken or less than perfect English, just because there is one guy in the room not able to speak German. Documents or code might be English, but the whole conversation around it is preferred in German.
This means when you don't speak German, you are mostly limited to work in projects where English is possible or the default - there are jobs like this, but this is actually limited.
I've met persons who were ten years in Germany and were not able to do a general conversation in a job interview in German. I would not hire them - other than for specific projects/tasks where that would not be a problem.
Basically nobody here will be able to avoid the healthcare system.