Do I have to prove German language skills when applying for a residence permit as a third-country national?

Yes. Aside from the Integration Agreement, third-country nationals who apply for a residence permit in Austria must prove German language proficiency for basic use of language according to level A1 of the Common European Framework of Reference for Languages (CEFR). ÖSD Zertifikat A1 and higher are recognised as proof of language competence in German. The certificate must not be older than one year at the time of the application.