I'm installing the green sheetrock (moisture-resistant) in my new bathroom. Is green mud and tape recommended or required also?
I haven't heard of green mud. For our bathroom, we used Densarmor instead of greenboard. The issue with moisture and sheetrock is the paper. Greenboard uses a waxed paper. Densarmor (and the like) use fiberglass instead of paper. I used the fiberglass sheetrock, fiberglass mesh tape, then standard joint compound. I then painted with a mildew resistant paint.
In the end, was it worth it? Well, it was probably overkill. Most mildew issues in a bathroom are the result of either a) direct water contact (shower area) or b) complete lack of ventilation.
When we did the bathroom, we also installed a high-quality ventilation fan (there was none previously) and a timer switch (to make sure it stayed on long enough after a shower). That, alone, was likely enough to prevent mildew issues.