Japan legally recognizes gay marriages (but only for those who marry foreigners in other countries)

The followings are just my guess.

I wonder what would happen if a gay couple did live openly in Japan:would they be treated well by their neighbors?

In Tokyo, people are not concerned about who lives next door.
And it is not uncommon that two men or two women cohabitate.People are not concerned whether they are gays or not.　People would usually assume that they are friends or relatives. Once it is known that they are gays, some people might be surprised, but that would be all.

Would they have the same job opportunities as Japanese heterosexuals?

I am not sure how it will really come out. I’ve read a news-report in which one gay person protested or sued against the company for firing him because he was a gay. The company said that he was fired on different grounds.
I remember a program on NHK that showed that a husband, who, I think , was a math teacher in a junior high school , came out as a gay. He had been married to a woman who was also a junior high school teacher. I think they had a child. The wife understood him. And his students at school also understood him. They continued the marriage, and he lives as a female-looking math teacher.

I can’t really tell whether a company will hire a person, knowing that he/she is a gay. I don’t think many companies will hire them as a receptionist or “an elevator girl”, at least for the time being.
There will be some troubles and some people might make fun of them at first, but the trouble will be the sort of troubles due to ignorance how to deal with them. As more and more gay people come out openly as gays, people will understand that they are, in a way, no different from the heterosexuals.
As I said, sure they have been marginalized in Japan but the point is that, as a poster on one of the links says, Japan “doesn’t have a religious reason to hate them”.
On TV, you see a sign that more and more people accept gays as such. (For some reasons, lesbians are rare on TV). They are entertainers but it is the first and important step for society to embrace them as just one of us.

When one of them died, would his/her partner be legally regarded as the inheritor?

Unless the marriage is legally recognized, it is impossible.But the person can draw up his will so that the property goes to his/her partner.

In the U.K., there have been instances where relatives, long estranged, have managed to claim inheritances, leaving their loved one’s long-term gay partner without a dime.

In Japan, that won’t happen.
If the person draws up his will to give everything he had to his partner when he dies, as a rule, the property goes to the partner.
His legal wife and children, for instance, “can” claim inheritance, but only up to the half of his property.

See Iryubun
http://ja.wikipedia.org/wiki/遺留分
http://www.ohtsukadai.com/iryu-bun/qa4-001.html