Name and rules of Japanese card game
Might it be Crazy Eight? (or Crates, Switch, Swedish Rummy, Last One, Rockaway, Mau-Mau or Pesten).

It has the most rules you mention, like reverse, draw two. Looks like UNO.

The last card rule might be a kind 'house rule', because there are a lot of variants of this card game all over the world.  

http://www.pagat.com/eights/crazy8s.html

In the japanese wikipedia article, its saying also Crazy Eights. Article translated by Google Translate.

https://translate.google.nl/translate?hl=nl&amp;sl=ja&amp;tl=en&amp;u=http%3A%2F%2Fja.wikipedia.org%2Fwiki%2F%E3%82%AF%E3%83%AC%E3%82%A4%E3%82%B8%E3%83%BC%E3%82%A8%E3%82%A4%E3%83%88