It doesn’t by default, this needs to be enabled in the options under Metadata > Genres. If you enable “Use genres from MusicBrainz” the rest of the defaults should be fine in most cases.
Please note that Picard currently is not really good at changing only a single tag, as it is meant to update your files with the data provided by MusicBrainz. If you don’t want Picard to touch other tags you need to add those tags you don’t want to get overwritten to the “Preserved tags” list in Options > Metadata.
Yes, you can not add an empty tag. Empty tags will not be written by default and you will just keep the existing data in the file.