I tried my first steps with drawing houses and housenumbers. Currently I am struggeling a little bit. The houses which were there before I started my editing have big housenumber on top. Means, when I zoom into a map I can see the housenumber on top of the house. When I am adding buildings and put attributes on the building, the house numbers do not appear afterwards
Looks like your edits are fine. And its nothing to do with using JOSM.
Its just how the standard map style on openstreetmap.org works. It only shows housenumbers for things which are not rendered with a symbol. So as that building is tagged as a fire station, it is shown with a fire station symbol (and the number isn’t shown). Or if you mapped something as a supermarket, it would be shown with a shop symbol, and without a number.
Whereas if you just map a house, it will just be shown with the housenumber or housename, and no symbol.
It is still worth adding house numbers for things, even if they don’t show up on the standard map. As other maps based on OSM might show the numbers. Plus they are very useful if anyone is searching for a specific house number and address.
Also note the map might take a while to update, so your edits might not appear immediately. You might be looking at a cached copy, doing a shift+refresh in your browser might help.