Yes. Especially if the patient has the above-mentioned "need to take something" because just waiting it out is not a sufficiently palatable option (even if it is the only one).

But IIRC you got to pay it yourself in many cases.

Not really. Some insurances offer to take the cost of homeopathy, but no doctor will prescribe "crystals" or other new age stuff. Are there black sheeps? Yes, like in every profession.

How a doctor reacts depends highly depends on the illness. If you have a cold, your doctor will recommend some tea and rest, but if you have a tonsillitis, you will get anti-biotics. And there is no such thing as a pain index.

Yes, they will. But to clarify: it’s not mandated or anything. It’s rather „good business practice“ aided by being covered by healthcare.

Almost never. You have to specifically search out those doctors.

Or avoid them.

Sadly, many Germans believe in homeopathy...