I'm not sure that I can add much beyond the excellent answers given by everyone above but, as a maths lecturer, my comments below might possibly add more detail about the PhD/research circumstances in maths.
First of all, you can definitely do maths research yourself and fulfill those four goals that you state without doing a PhD or working within any institution. Grab a good maths book like A Course in Combinatorics by Van Lint and Wilson (sorry for my bias!
) or any other maths book that might appeal to you, browse through it until you find some topic of interest, and enjoy the content and the challenges of filling in the intentional left-to-the-reader gaps and the problems, while at the same time wondering how you could possibly generalise or modify mathematical results to become something larger or new. Webpages like
www.cut-the-knot.org and ProjectEuler.net are awesome sources of fun and inspiring problems and potential research ideas and projects, and there are journals such as Parabola (
www.parabola.unsw.edu.au) that publish papers by non-professional mathematicians. If you see any articles that you like, then you could contact the authors and see whether any happy correspondence, or even collaboration, might happen. That happens more often than one might think.
If you would like to do more "professional" maths, then the suggestions above of asking professors for "summer project" type research projects are very good. Many professors are able to provide you with these, and many professors are eager to help you, and are grateful for help in their research. As mentioned above, it's a bit of a gamble for a professor to take you on in any supervisory capacity, but if you tell them that you already have a maths degree and are very independent, not asking time or effort from them, then most professors would be happy to help you. That is, if they have projects lying around and if they aren't too busy to answer their emails(!). Unfortunately, I (Thomas Britz, UNSW Sydney) don't have any good research projects to offer myself right now; otherwise, you'd be more than welcome to look at them! (You're also more than welcome to email me about these issues or anything else!)
Most pure maths doesn't require equipment. I sometimes do large-scale computational searches using university computer clusters but if you are good with computers and can figure out how to parallel program GPUs, then you can do these sort of computations with a $200 graphics card. A friend of mine does that.
A bigger obstacle to maths research is literature, at least if you want to work with cutting edge research (which you might not! Cutting edge doesn't necessarily mean fun.) Pure mathematicians were the first researchers to introduce and embrace open access journals but most maths research papers are still inaccessible to people without institutional access (unless you're willing to pay a fortune). There are ways around that, and sites like arXiv.org help, as do the increasing number of open access journals. You're also more than welcome to ask me for any articles too!
As for the quals, they are mostly to vet you: it's a gamble to take on the PhD studies for the student, the supervisor and the institution, and if a student's maths skills or motivation aren't excellent, then it's a lot of pain for everyone, especially the poor student.
With respect to background knowledge, it usually helps a lot to have some, but this depends a lot on the area of mathematical research. You might need 4-5 years of uni studies in order to understand questions about cohomologies, say, whereas many problems in graph theory can be explained to school kids. On the other hand, the answers to those cohomology questions might possibly be a lot easier to find, given that the questions are very specific and that far fewer people are looking at them. The importance of background studies - and those quals - is maybe more that you've trained yourself to top mathematical "fitness", and already have a diverse array of general tools to use. It might be a struggle to return to studies and to learn from scratch, but it's definitely possible and I've supervised thesis students who were old and retired from their careers.
As a PhD student working independently from at home, the biggest challenges are usually lack of motivation, inspiration and loneliness (funny enough, I was mostly in that situation and loved it, finding it to be free, creative and happy - but this is probably atypical).
If you're doing independent maths research for fun and personal development, then this is not that much of a problem, and you can find likeminded people to chat and perhaps collaborate with.
However, if you want to publish research journal articles, then it might be hard to find projects to work on, to get good ideas and feedback on how to find new results, to be able to look through the literature to see whether your new results are actually new, and to know how to write up and present your results. StackOverflow is a good source of good research ideas and feedback for non-professional mathematicians, and zbmath.org is a free and excellent database for searching maths articles. Still, the PhD studies provide a lot of cultural training, including how to write a maths paper clearly and presentably, that is hard to get elsewhere.
I mostly worked on my own during my PhD project and had little supervision and next to no feedback from supervisors. That made me work hard and independently but I missed out on collaboration and support that you get from a group and its networks. In the other extreme, some people are so supported by their group that they don't learn to be independent. Most maths is usually collaborative, though often in parallel or turn-based. Some people work more directly together; it depends on people's dynamics and the work at hand, and the modes of collaboration can be quite diverse. It's usually constructive and collegial in maths, though, often socially boring but also without unnecessary frictions or politics.
I hope that these comments added something to the discussion and that you might find them useful!
If you're interested, I've written more about related topics here:
https://www.quora.com/profile/Thomas-Britz
Oh, and good luck and enjoy your maths research, in whatever form it might take!