I'd like to have a software that allows me to download and view (OpenStreetMap-) maps of the whole world or a certain area I chose to download. Ideally, it should behave just like the map-website but be usable even without internet connection. Do you know such an app?
Try GMapCatcher https://code.google.com/p/gmapcatcher/. It has support for downloading maps from various providers, including OpenStreetMaps. I found it to be quite fast even when browsing online, as compared to the web version of google maps.
It has lots of options (https://code.google.com/p/gmapcatcher/wiki/User_Guide). I have not tested all of them, but it seems to be promising.