I'm happy to see this question on our Meta site and I like the idea of incorporating the site into classroom activities. I'm not an educator but I would like to give a bit of a reaction from the perspective of what the site is looking for, in terms of what types of activities might be more productive vs. more annoying for our users.
I would not generally recommend any activity that imposes a pressure on students to ask any number of questions on the site. The bulk of our community moderation work is tied up in reading, assessing, responding to and sometimes closing questions from new users. Figuring out exactly what sort of question works best on Stack Exchange can be a very trying experience, both for the user who's figuring it out and for the experienced users who are interacting with that user along the way.
So, for example, "ask a question on Engineering SE during the term and I'll give you some number of extra credit points" is something I would recommend against. Assigning "ask a question" as required homework would be even less productive from our perspective; probably most of those questions would not be well-received.
Encouraging students to answer questions, on the other hand, seems fine because the workload involved for us in maintaining answers is much lighter. Mainly because we don't close answers, but also because everyone has their own expertise and can't evaluate every single answer.
You could also set up a scenario for your students that puts them in the role of a professional who comes across Stack Exchange for the first time when trying to solve a problem. I'm imagining something like this:
You are a junior employee at ABC Chemical and you are assigned a task by your supervisor. (Description of task here.)
You searched the internet and found a page on Engineering SE that might offer some important information to help complete this task, or maybe even a full solution. (Link to Q&A here.)
Read the question and its answers. Assess whether you think you can solve the problem using the information on the page. Write an email/memo to your supervisor explaining what information you found and how you're using it to complete the task you were assigned (if at all).
This would likely involve some additional research on the student's part, whether it's following links directly from an answer, trying to verify an answer that sounds good but has no citations, or deciding that the answers there are all crap and they just need to figure it out for themselves. If in the process, the student decides they can submit a productive edit, wants to sign up and vote on the posts, or even finds they can write their own useful answer, those would be welcome extras.
Definitely an interesting application to think about in any case, and I'll be very interested to read the thoughts and experiences of other educators in response to your question.