While I think some of your questions point to genuine weaknesses in math departments (a lack of an internship pipeline, for example), a lot of them reflect a really unrealistic idea of what "cooperation" means. Your DGS, chair, etc. have met a lot of 1st/2nd year graduate students, and thus they know that very few of them are ready to do serious research, and in fact very few of them really know what that would mean. It wouldn't be looking after your interests to leave you to do research on your own, with no way of checking that you aren't just wasting your time. 1st year classes exist to cover basic material that any mathematician should know. If you know the material, it would have been reasonable to skip them (and usually departments will give students flexibility to do this), but as I say below, if they are taking up all your time, that gives the lie to your suggestion that you were already ready to do research.

How common is it for math students to be supported by external funding (e.g. NSF GRFP) rather than a TAship?

Rare, outside the very top schools.

How common is it for math students to do internships in industry to gain practical experience?

Rare in my experience in pure math departments.

How common is it for math students to do outreach or volunteer work during graduate school?

More common. I've known lots of people who taught (for example) in prisons as volunteers, or worked with K-12 students. Depends a lot on what's available.

Does a math department generally provide support and guidance to students when it comes to finding and applying for fellowships and internships?

For fellowships, they should but the reality is more mixed. In my experience, the problem is more that students aren't willing to go through the application process. It might require a little initiative to get help with this.

Is it a common attitude that it is the student's responsibility, if he wants those things, to undertake all steps of this process by himself?

Yes. Graduate students are adults, and responsible for themselves. It might happen that someone in the department is looking for graduate students for an opportunity like this, but at the end of the day, it is up to you.

Is it normal for math departments to disregard students' research interests in favor of mandatory coursework?

This is pretty insulting: obviously the department requires the coursework because they believe it is in the students' interests. Of course, it's hard to have a system that fits everyone, but very few students could be successful in grad school in math without taking a couple of years of classes (again, with the possible exception of the best schools).

For example, because all of my time is tied up in the 1st/2nd year courses and exams, I have been unable to do any research for the last two years, despite being very prepared and capable on my first day. (I published during undergrad and was chasing several promising ideas when I arrived.)

If you were very prepared and capable, then your first year classes should have been easy, and you should have had plenty of time. If they were hard enough to take up all your time, you didn't know the material.

When I asked to be allowed time to do research instead of taking classes, I was rudely shut down.

I can't comment on whether it was rude or not, but I can't say it was a surprise that this approach did not work. Since you haven't mentioned a research advisor, I'm going to assume you don't have one. Very few students would be capable of doing this, and I have trouble imagining that your undergrad publications were so strong and independent that the department should have had faith that you were one of them.

How often do mathematics departments get together in social or community events, such as departmental happy hours?

Depends. Departmental happy hours aren't a strong tradition, but department teas are, and usually departments have a department-wide social event like a picnic or holiday party every semester or so.

How is networking different in mathematics? Do professors generally have connections in industry or prominent members of their field, or is that a rare thing?

Rarely in industry, usually with prominent members of their field, but this depends a lot on the definition of "prominent" and "field."