I often observe from Bortle 3 or 4 skies with scopes and find myself looking up for hours on end. Here are my thoughts on your questions.
1. The Milky Way is literally an important part of the definition of the actual Bortle scale. Thus, in order for a sky to be Bortle 6, you need to be able to see the Milky Way near the zenith. Now, assuming you are new to the hobby, you may not realize that most light pollution maps are garbage. They use old, obsolete data and are generally based on modelling. Thus, what many maps say is Bortle 6, is probably closer to Bortle 7 or 8 most nights. Transparency also can vary due to humidity, smoke, dust and clouds on a nearly hourly basis.
All that said, while the textbook answer for the Bortle scale needed to see the Milky Way and Zodiacal Light are Bortle 6 and Bortle 4 respectively, if you are using light pollution maps, the skies probably need to be at least one or two levels better to account for general degradation of the skies over the last few years. Experience matters too. I can sometimes see the Milky Way from my house, which is Bortle 7 most nights, but I know exactly where and when to look.
2. I don't have a good answer to this. Bortle 4 skies have many more stars than I can count (literally thousands) and I've never even bothered to try to count the number visible from my Bortle 7 sky. When I lived in the Washington DC suburbs (Bortle 9 or worse), there were none except maybe Sirius, Vega and Arcturus, and even then you needed to know where to look.
3. They will be pale white/gray, just like almost everything else except the occasional bright star or the aurora. Your eyes are not very sensitive to color in the dark. Some subtle shading is visible to seasoned observers, but we are literally talking about shades of gray. Structure, including brighter and dimmer areas, will be visible, but color won't be.
By no means am I trying to discourage you, but it is important that you have realistic expectations. Your eyes can't take long exposures, so they cannot hope to match what is visible in photographs.