Does one take on duties from the previous owner or previous resident?

The hedge was tolerated by the municipality, but it was still the property of the "previous owner." You have now bought it along, quite simple. The obligations pass on to you.

The hedge is on municipal land. He bought a property with a house and everything that belongs to it, the hedge on municipal land is not mentioned in the land register. I am pretty sure it is not his problem. There simply is no legal basis. If the municipality doesn’t like it, they can remove the hedge, he doesn’t need it.

The hedge is on municipal land.

Ownership often isn't about what something stands on, but who it belongs to.
Glad to see and understand in boundary construction.
If I plant a tree on your land, it is not yours, but mine.

If I plant a tree on your land, it is not yours, but mine

But if you don't pay me rent for the location of your tree on my land, then it will take a while until my tolerance becomes your customary right or until I need a felling permit from you on my land (?)

There is no customary law. Whether you get a felling permit is a completely different matter. It is about mine and yours.

There is no customary law.

Yes, usually after 100 years – that is where the limitation of hereditary leases to 99 years (without extension) comes from. And we have already had the topic here several times about appeal periods expiring after certain toleration durations, especially with "shared" enclosures (unfortunately I cannot provide "as usual" links for this).

Whether you get a permit for felling is quite a different matter.

I did not mean from the municipality as the issuer of the tree protection statute here, but in your example from you as the condition violator with your tree on my property.

Well, only if it is stated like that in the land register.
Is that the case?

Basically, I think it’s a fair thing.
You have the privacy screen from the hedge without it standing on your property.
It’s reasonable to trim it twice a year with the hedge trimmer.
Picking up and disposal is probably the most work.

Not necessarily. If the previous owner signed something for themselves and all successors, the obligations were sold along with it. It’s not that simple.