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…

There is no such thing like EU citizenship. All countries have still some kind of independence and own laws. As far as I know nobody can be forced to vote, and in almost all countries there is no jury duty except of France, Austria and Belgium(?). But I am not sure about that.
there is in France? it must be rare as I’ve never heard anyone had to be jury
Juré d'assises

Tout citoyen français qui répond à certaines conditions peut être juré devant la cour d'assises

Of course there is something like EU citizenship. Read the TEU and TFEU. It is conferring its own set of rights but is dependent on someone being a citizen of an EU member state. Hovever there are cases were people who are the one aren’t the other.
There is no state of EU therefore there is no citizenship. You just have the citizenship of a member state.

You can talk about national citizenship all day if you want, that doesn’t change the fact that EU citizenship exists, comes with meaningful rights attached to it, even if it is not a national citizenship. Mind you, you did not say, that EU citizenship is no national citizenship, you said that EU citizenship does not exist which is demonstratively false.

Article 20 TFEU

“1. Citizenship of the Union is hereby established. Every person holding the nationality of a Member State shall be a citizen of the Union. Citizenship of the Union shall be additional to and not replace national citizenship.

  • Citizens of the Union shall enjoy the rights and be subject to the duties provided for in the Treaties. They shall have, inter alia:
  • (a) the right to move and reside freely within the territory of the Member States;

    (b) the right to vote and to stand as candidates in elections to the European Parliament and in municipal elections in their Member State of residence, under the same conditions as nationals of that State;

    © the right to enjoy, in the territory of a third country in which the Member State of which they are nationals is not represented, the protection of the diplomatic and consular authorities of any Member State on the same conditions as the nationals of that State;

    (d) the right to petition the European Parliament, to apply to the European Ombudsman, and to address the institutions and advisory bodies of the Union in any of the Treaty languages and to obtain a reply in the same language.

    These rights shall be exercised in accordance with the conditions and limits defined by the Treaties and by the measures adopted thereunder.”