Negative COVID-test required during immigration at a Schengen layover or by the airline?

TL;DR: Austria has waved off the requirement of negative COVID-test for some specific cases including mine. Traveling from India to Austria via a layover in another country in Schengen. Will the negative COVID-test still be required by 1) immgiration at the layover (another Schengen country) or 2) the airlines?

I am currently in India and hold an Indian passport. I am in the process of applying for visa D to visit my husband in Austria. My husband holds an Austrian residence permit (Rot Weiss Rot karte plus). Austria usually requires a negative COVID-test or home quarantine for people coming in. However, on the official Austrian government ministry webpage (answer to question 5 in the link), they have clearly mentioned that visits for special family reasons (which includes visiting a spouse) are exempt from both of the above requirements.

Almost all the convenient flights from India to Vienna have a layover in another city in the Schengen area say Frankfurt. Since I will be entering the Schengen area at Frankfurt in that case, I suppose I will have to go through immigration there rather than in Vienna. Will I be required to present a negative COVID-test at the immigration in Frankfurt even if I am there for a short layover(a couple of hours)?

And secondly, are the airlines too asking for negative COVID-test before letting the passengers board? I will mostly be using Lufthansa and Austrian, so the answer could be specific for only these two.

I tried looking through the websites of the German government and the airlines too, but I couldn’t find clear answers. Any assistance would be greatly appreciated.

Thank you.