Munich to Salzburg Round Trip - Blue Card

A partial answer regarding your third bullet point:

Quarantine rules on return to Germany will depend on how Germany classifies the pandemic situation in Austria, notably regarding mutant varieties (Virusvarianten-Gebiete). This classification may change on short notice, even while you're already abroad. So better plan with the worst-case scenario of 14 days quarantine for a resident of Bavaria, and take it as bonus if you can get out earlier.

The German government strongly advises citizens and residents against travel to Austria. They don't do that lightly.

The rules in Austria will radically change on the 19th of May.

For the immigration side of the question, as a resident of Germany you may freely enter Austria and return to Germany.

At the moment there are about 32 daily train connections between the 2 cities.

Covid-19 conditions when returning to Germany will be based on whether you are fully vaccinated (or recovered) or not.

A test (and quarantine) is presenty not required for those who are fully vaccinated (or recovered).

The entry rules for Austria (starting on the 19th) have not yet been finalized, but it has been announced that vaccinated (or recovered) do not require a test or quarantine.
Note: the Austrian text does not use the word fully, so this is a bit ambiguous.

Inside Austria:

For visiting restaurants, hotels, culture and sport events, no test will be required for persons 22 days after they have recieved the 1st vaccination or have recovered.

This law has been finalized and comes into effect on the 19th of May.

You don't write why you are going to Austria, but going for tourism on May 14th will not be possible.

Coming from Germany, you are currently after entering Austria required to quarantine for at least 5 days unless you are covered by any of the exceptions, which you as a tourist will not be. Hotels and restaurants are also closed.

New rules in Austria will take effect on May 19th.