Generally asking multiple questions in a single question are discouraged on Stack Exchange sites, since asking questions in bulk reduces their exposure as individual questions.
@jakebeal is correct, authorship and the ordering thereof, is extremely field-dependent. A tag search on Academia.SE should give you an idea about how different things are in different fields.
Here are some general answers that might hopefully be of some use:
- Do advisory committee given a different treatment when it comes to the contribution? Are students expected to list them as authors?
Co-authors are people who have contributed to the work at hand. How that's defined is quite up to the field and sometimes even specific groups. I know for instance that in maths and physics, even having a lengthy conversation over a cup of coffee or two might be rewarded with a co-authorship. It is somewhat more strict closer you get to biology and medicine. But there is no written rule as far as I know..
- How are contribution and authorship communicated, what is the norm and when draft is sent or circulated through email (between members)
should the student add their names as authors, leave it blank, so
whoever want to contribute will add his own name?
In my experience, it's usually (and hopefully) decided in advance and revised prior to submission or after reviews, in case any extra work is required. I have never heard of "sign up if you wanna be a part of this article" approach before.
- Should the PI providing the data or equipment be listed on the students work even if he doesn't light or contribute to the drafting
them a script?
There's a long, and I mean long path between drafting a manuscript and getting it published. The PI will most likely be involved in those steps in between, if not in the beginning.
You also need to see that the role a PI assumes might vary from person to person, some are hands-on and some adopt a more sink-or-swim approach. Some will micromanage details, and some will leave it up to you to figure things out along the way. My point is, if your PI hasn't been contributing much so far, there might be more than one plausible explanation for it, don't be quick to judge.
And finally to answer the question, I cannot imagine a world where you can exclude the PI who's providing your salary, and the resource you've used from a publication that you have produced, unless the person specifically tells you to do so.