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.

Same as most European countries, Germany has the adversarial system - no jury, just the judge.

Don’t know about elsewhere, but in Germany, for some trials, there are an additional 1 or 2 “lay judges” - non-professional judges from the community.

Quite similar here in Czech Republic. No jury duty, because there is no jury and voting is absolutely up to you.
These lay judges are the remnants of the jury system, they are basically a mini jury.
There is a form of jury duty in Germany: Schöffe (honory judge).
[Schöffen] (de.wikipedia.org/…/Schöffe_(ehrenamtlicher_Richte…) duty can be mandatory, if not enough volunteers can be found (or are not deemed trustworthy enough by the local council).
Schöffe (ehrenamtlicher Richter) – Wikipedia

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