- how foreigners see Japanese women as being easy, and the fact that there are women that live up to the reputation
This doesn't actually come from foreign men that live in Japan, but more from the media (both Japanese and Western) by creating sensational mythology to sell books and from politicians appealing to xenophobia.
Then there is a bit of catty jealousy from the opposite sex. That is foreign/Western women upset with foreign/Western men dating Asian or Japanese women. And Japanese or Asian men upset with Japanese/Asian women dating foreign/Western men.
There are many Japanese women that are afraid of foreigners, believing the most negative stereotypes, like they are criminals, rapists, or all playboys. They may overreact, like think foreign men say they are yellow cabs and other such ridiculous non-sense. The paranoia of too many Japanese women can be extreme. Western guys in Japan, are usually not, calling Japanese women yellow cabs. This is often foreigners that have NEVER been to Japan, or have NEVER met native Japanese people, and repeating something in the media. Often this is not directed at just Japanese women, Asian women in general. And the other culprit is often Japanese guys, being jealous or territorial. These type of guys don't see it a problem for them to lust after or date White or Western women, but will try to control all Japanese women like they are their personal propery or not allowed choice.
The reality is Japanese women are very difficult for many foreign men. They can't speak the language, different culture, don't understand the body language, don't have the chance to meet them (not from the neighborhood, college, or same job).
I had a couple of foreign friends upon arriving to Japan, that did not have a date for their first 6 MONTHS. My other friends and I had to explain to these guys how to talk to and date Japanese women. And it was difficult helping those guys. In no way are Japanese women "easy", actually they are more or way too difficult.