Interesting, because the only time I think in a language is when there's speech in my thoughts. The rest of the time it's thinking in images or things just happening. So that ties in with thinking in sign language, and then thinking in another language when you've been speaking another one for a day or two.
It's definitely Japanglish: Japanese plus any variant of English. It doesn't always come out in words as nicely as I'd like but it works. I've been known to speak well in other languages and often without the translation layer but I don't feel as though I'm thinking in that language really.
When I lived in Japan, I would think some things in Japanese, some in English (most English). Having been back in the U.S. for a while now I think in English, but still occasionally find myself thinking something in Japanese. There are a few things that are just easier to express or more natural in Japanese I guess...