Can I install Android app without all required app permissions?

If an app targets Android 6 or higher, most of the permissions listed in your screenshot will have to be approved (or denied) by the user at runtime – i.e. when you're using the app. There are several permissions not falling into this category, as Google defined them as "normal" (and only "dangerous" permissions are considered worthwhile to be presented to the user). So you e.g. will never be asked for Internet access (if you denied, Google couldn't show ads).

OTOH one of course could argue: if you're concerned about your privacy, Facebook is the wrong place to go in general :) And of course again, if you don't trust an app, watch out for alternatives. Best at F-Droid where you can trust every app.