There's also a very large gay entertainment district in Tokyo (Shinjuku Ni-chōme ~ 新宿二丁目).
LanguageThe common Japanese word for gay is homo. Don't be offended — it doesn't have a negative connotation.
StatusThere is no law in Japan against homosexuality but Japan doesn't recognize gay marriages. It hasn't been a big issue in Japan (yet) so many Japanese may be unaware there's any such thing as gay marriage.
You're unlikely to convince a hotel to give you the honeymoon discount but there will be no problem staying together.
Love HotelsLove hotels do not allow lone men or male couples to stay. The one exception may be hotels in Shinjuku Ni-chōme (not confirmed). Groups of women are generally allowed to stay at love hotels.