Are both of these things mandatory for EU citizens?

I'm mainly talking about: - Voting (at federal elections) - Jury Duty (at a courtroom) I know that in certain countries like Australia f…

Speaking for germany: no mandatory voting, no jury duty.

Being part of a jury is voluntary and you have to apply for it. It’s also probably quite different from the US jury system.

You can however called up to be a Schöffe: de.wikipedia.org/wiki/Schöffengericht (no English version available, your friendly LLM next door should help out)
Schöffengericht – Wikipedia