There are several threads here already about adding only the genre tag or only a single tag in genera. Have a look at:
Regarding the naming you can use the %genre% variable in your file naming script to place the genre information in the file name.
Note that there can be multiple genres and that genres can be per track, so probably not a good idea to use genres as part of the folder name. But using it as part of the file name should be ok then.
If you want to use only a single genre you can limit the number of genres Picard loads from MB in the options (Metadata > Genres). If you limit this to one genre Picard will only load the most upvoted genre the loaded track or release. There is currently no good way to get only the most generic genres (such as pop, rock, jazz, metal) if that is what you are looking for.