Regarding the nitrogen: I heard otherwise, that composting it in-situ will reduce the available nitrogen and you should add some long-lasting nitrogen source to it, like e.g. horn shavings

There’s some truth to that, but it’s a bit more nuanced: It depends on what you’re composting. With substrates that are very high in carbon and low in nitrogen (such as wood chips or shredded wood, if you have your own chipper), the low nitrogen-to-carbon ratio causes the microorganisms that break down the wood (fungi, bacteria, etc.) to draw the nitrogen needed for cell growth from the soil—and in that case, you’re right: As the mulch decomposes, it initially leaches nitrogen from the soil.

The situation is different when using very nitrogen-rich material like fresh grass clippings: These contain a relatively high amount of nitrogen compared to their carbon content, so the microbes don’t need to draw nitrogen from the soil beneath them… In that case, you have to be careful not to make the mulch layer too thick, as the combination of high metabolic activity and high moisture can create oxygen-deprived conditions (this is why a pile of grass clippings becomes wet, rotten, and mushy and no longer decomposes properly). In my experience, a 10-cm layer of fresh grass clippings as mulch on my soil is no problem and breaks down well.