I think you shouldnt have any Problem getting a Working holiday Visa (if that WHV agreement is also for Denmark.
It should also be possible to find a Job as English teacher if you are not native.
Not everyone in tokyo speaks english, but most people who want to teach come here, so theres alot of competition (at least more than a t other places).
If you want to improve your Japanese try not to work as a English teacher as you will use english all day and most likely also talk in English to your workmates. If you have a working holiday Visa you can get other jobs wich will probably not pay as much as teaching, but enough to make a living (if you dont spend a couple of hundres a month on partys or stuff).
I currently live in a Guesthouse in Tokyo and have around 1 hour commuting time to work and i spend around 5000-6000 yen a day (currently) but hope to drop that figure a bit later this month. Around 2000 for the room (got a good deal) around 200 for Transport (got a monthly Ticket) and around another 1500 to 2000 for food. The rest depends on how fancy you want to live (alkohol, cigarettes, party,....).
Just had a quick google check, the WHV Japan-Denmark exists (i didnt know) since recently (around a year ago) so i think you should be fine with that.
Oh one more thing: Depending on what kind of Japanese you want to learn, it would maybe be wise to visit a school at least for sometime. What you will learn by living here will be spoken everyday language and maybe not the politest form. If you want textbook Japanese then go to a Language school in Japan.
Anyway, good luck in your plans, Japan is amazing and im happy there is a WHV for Germans
(but ive also had some hard times already...)