@whynotsim None of the three computers you've listed are anywhere close to as fast as the Skytech Azure I linked, so if you're comfortable in that price range, the Azure is the best bet. The first one you listed looks fine to me and is a good price, but I don't know anything about this particular seller, either the build quality of its systems or what kind of support you'd get. I would consider it somewhat of a risk just because buying from a third party through Newegg, as opposed to buying something sold directly by Newegg, is always a risk in terms of support options. On the other hand, if the computer works as advertised, then you won't need support and so that wouldn't be an issue.
The second computer you've listed is the same price as the Azure and not nearly as good, so I'd just cross it off. The third is basically a laptop in desktop form, without even a dedicated graphics card. It's a good option at that price but is absolutely not a gaming PC and shouldn't be advertised as one. Even a desktop 1650 is more than twice as fast as the integrated graphics chip in this computer.
There are two potential problems with mini PCs or small form factor builds, by which I don't mean the third option you linked but rather the Skytech Chronos and others like it. The first problem is that it can be difficult to work inside such a small space, especially for people who aren't already comfortable building or upgrading computers. That would of course only be an issue if you were looking to upgrade. The other problem is cooling: it's more difficult to move air out of the case fast enough to keep the components cool. This shouldn't be a problem with a properly-done build or with the components in the Chronos, which wouldn't generate a lot of heat. So I wouldn't tell you not to buy one in this price range.
Almost all new-ish monitors work with almost all new-ish computers. DisplayPort and HDMI are standard connections, with any given graphics card from the last five years at least supporting both, and most monitors supporting both as well. It's really just plug and play here. However, the iMac monitor wouldn't be compatible out of the box, so to speak, and it might not work even with some advanced trickery. I'd have to look into it more, but I can tell you for sure that this kind of setup is not supported.
In terms of what monitor you'd want, the first question is how large you'd like it to be. Then you can get into resolution and refresh rate. The other critical detail is that you ideally want IPS, the highest-quality panel in terms of colors and viewing angles. VA is good enough if you play in a dark room and a reasonable way to save money, especially if you're getting a monitor with a higher resolution or refresh rate. But like I said, choose the physical size first, then we can go from there.
