[ファブリス]

Is it possible to send emails directly to someone's mobile phone in Japan?

Yes. Well, I'm presuming as I have sent e-mail to my own phone there. Your carrier may vary, but I doubt it. The part I wonder about is the "quoted" portion that seems to get snipped in transit.

if they have an address like jjfajf@docomo.ne.jp or @ezweb.ne.jp or whatever it should work.

I often send emails between my gmail account and my AU phone

should be ok since I send emails to myself all the time BUt there might be some issues with encoding from pc to phone.

Yes, I routinely send mail from a standard email account to a phone in Japan. However, there are some limitations. I believe there is a size limit. And I have to use plain text... no colors, no fancy fonts, etc.

Some phones can only receive 500 half-width or 250 full-width (or any combination, with 1 full=2 half). Most can receive more. I've set my phone to recieve up to 1500 characters in 3 chunks of 500, but the initial setting was 500 max.
Other than that, Japanese phones can send and receive e-mail with any other e-mail equipped device.

It will work, but you need to specify the encoding properly (iso2022 I believe) otherwise it'll be 文字化け. Many users also block emails from computers to stop 迷惑メール (spam), in which case nothing you do will get the email through until they turn off the filter in their settings.

I send utf-8 encoded e-mails to a phone in Japan regularly.

Every utf8 mail I've ever sent turned into mojibake. Maybe some phones can do utf8, but it's better to be safe.