Is there a website or any kind of document that shows all currently airing series in Japan? Airing dates and times would be great too.
There's AnimeCalendar, which, when you make an account and are logged in, lets you filter the series and also convert to your local timezone.
There's a MAL Group that updates with the season, has airing dates for all the anime airing during that time and such. It's right here.
I always use Mahou.org, which not only tells you what is airing, but gives you up-to-the-minute information on precisely when it is airing. However, you'll need to cross-check the listed shows with some other site like MAL, AniDB (note the convenient links to the right of the table), or ANN's Encyclopedia if you want descriptions.
There is also the Fansub Wiki which separates out each season, includes the time/date/channel, is updated (at least regularly enough these days) to show which shows get picked up by which groups that aren't licensed by U.S. distributors. Crunchyroll, Funimation, and Sentai pick up large handfuls of shows each season these days so the actual fansub groups are less relevant now.
I just noticed today that ANN has an upcoming anime list.
Neregate also has charts of all animes, OVAs/movies/whatever else will be airing the next season. (So you can check what was the "next season" of last season, i.e. this season.)
Note that while the charts on Neregate are family-friendly, other elements of the website may contain NSFW content.
Wikipedia has listings of anime per year. For example: Anime Television Series of 2012.