Isn't Russian your first language, anyway, and English is your every-day-of-your-life language?  When I think "bilingual", people like you are what I think about.

Fluency in ASL is the same as any other language -- thinking in it, dreaming in it, and so forth.  I HAVE found myself thinking in ASL once when everything clicked in class, but that was only once, and I could only form very simple thoughts.