How many questions are asked on different topics?
You can look at the tag overview page to see how many questions are tagged with which tag. That should give you a rough idea under the assumption that questions are tagged correctly. Obviously not every question will always have the right tag and it depends on whether the granularity is enough for you.
How many of them are answered and/or commented?
If you go to the questions overview page you can see that as of writing this Mi Yodeya has 25,829 questions. By clicking on the Unanswered tab you can see how many of these have not received an answer. You can look at those that have received an answer, but none of the answers have a positive vote count and those without any answers. Right now there are 3,246 questions without any answer. This translates to roughly 12.6% of questions being completely unanswered.
How many of them are accepted answers?
You can use the search bar with the advanced search (a link on the right side gives you more options) hasaccepted:yes to find all questions with an accepted answer. Right now there are 8,851 questions with an accepted answer, which is roughly 34% of all questions on the site.
How many users contribute to what portion of all discussions
This number is more difficult to find and depends on what you mean with "discussion". For example here on Meta there is the discussion tag. Or do you mean answers? Posts in total, which would include questions? Do you include comments?
You can use the StackExchange Data Explorer to get some numbers if you know what you are searching for. For example the query:
SELECT Count(*)
FROM Posts
WHERE OwnerUserId = '28789'
AND PostTypeId = 2;
would give you an overview over all answers I have created on the site you are searching for. You can find your ID by going to your profile. Mine for example reads https://worldbuilding.meta.stackexchange.com/users/28789/secespitus and my Id is 28789. Just choose the ID of someone you are interested in and plug it into the query to get the amount of answers they have written.
Of course this little bit of information is also available from their profile page, so you would have to write a far more extensive query. But this may get you started with your search.
Can the users be distributed by their ethnical/community affiliation, learning experience, country, sex, age etc?
No, such information is not available. You could look at the profile and see whether someone has written something like that, but in general such information is not available and it's up to anyones guess.
Question distribution by Halahic topics of Shu"A (Ora"H, Yo"D, Cho"M, Eva"E)?
I am just passing through and randomly found this question, so I don't understand this point. Sorry.
Rabbis and books cited?
With the SEDE you can analyze all answers on the site for certain keywords for example. Maybe that helps you to get those numbers?
