Mask requirements in Germany: public transportation, museums, restaurants?

You can find current info. on the official website for Germany.

https://www.auswaertiges-amt.de/en/coronavirus/2317268

Mouth and nose must be covered aboard any public transport. Masks must
fulfill the requirements of FFP2.

Measures for fighting the spread of COVID-19 are subject to local
regulation.

Some museums are requiring masks, also, even though they are not required to by law.

Each state has different regulations.
None of the museums in Hessen require masks unless you are also on a tour with a guide in the museum. That seems to make a difference. Masks on all public transportation is Germany wide, but if not mistaken, Bavaria requires FP2 masks. No cloth masks anywhere. Only medical, but they are widely available everywhere.

You can also keep up to date by reading the Local or Deutsche Welle.

FWIW, N95 & KN95 are the equivalent of FFP2. Does anyone know if those types are accepted in Germany, or is there an issue if you don't have a FFP2-branded mask?

No issues. I don't think anyone is really checking to be honest. They are all similar in style.

I wore a KN95 mask the entire time I was in Germany and no one said a word. In fact, I saw many Germans wearing the same mask.

Just got back from three weeks in Germany. If it moves (train, bus, boat, plane) you need a mask. Compliance was near 100% except for some big city subways.

Definitely needed them to enter an apotheke (pharmacy).

Group tours in the Bavarian castles (including Neuschwanstein) also required them.

Although it's still recommended at most museums and indoor locations, I can't think of anywhere else it was required.

While there this month (July) masking was regularly announced on the ICE train’s PA system as required. Other than that rarely did I encounter mask enforcement even when signs were posted requesting such. I simply wore my mask regularly.

We returned just this week. We were required to wear one to tour inside at Burg Eltz castle. In Munich, signage at the Residenz and Alte Pinakothek museums said masks were recommended, but hardly anyone working wore one. In Cologne, our English tour guide wore one, so we did, as well, out of respect. The rest of our group did not, as masks were "recommended". In Boppard, we took a Sesselbahn chair lift over the Rhine river valley, and had to put on a mask in order to board the chair lift (and save it for exiting the lift).
On Thalys, as we came into Germany from Belgium, they announced masks would be required. About 75% of people put one on, though the train operators said nothing to the others. On the busses in Cologne and Munich, everyone wore a mask, besides the driver who was behind plexi glass. You'd stand out like a sore thumb without one (like us when we first hopped on and were more worried about buying our ticket than putting on a mask, oops!) On DB ICE train, the recording came on to remind us to wear masks, but many people did not. I did not see any operators talk to people about it.
On Condor airlines on the way home, people were reminded individually by flight attendants to put their masks back on when no longer eating or drinking. No requirements in hotels, shops, or restaurants that we encountered. We took a cruise on the Rhine on KD line, and no one said anything when we forgot to put ours on.

We kept spare masks with us at all times, so that we weren't caught in a pinch.

KN95, N95 (or similar), or surgical-style paper masks (that unfold accordion-style) are all good. Cloth masks do not meet the mask requirements in Germany. No one is checking, and we spent all of our time in Bavaria, so nothing special there.