Managing macOS application windows like Ubuntu Unity
I'll take your points in order:


You can assign entire applications to desktops, but I don't know of any tool that can assign specific windows to desktops, at least out of the box. Maybe you could whip something up with Hammerspoon and Lua scripting.
I've never actually used Hammerspoon, I've just heard about it. If you decide to go this route, good luck!
That's not actually how you move windows between desktops. That dock menu is for the entire application, not the specific window. (Windows actually don't even have window management menus at all!) To do this, grab the title bar and press ⌃← / ⌃→ or do a 3- or 4-finger trackpad swipe, or hold the window at the edge of the screen for a moment. You can also move windows between desktops with "Mission Control", a combination of Expo and Scale. You get to that with ⌃↑ or a 3- or 4-finger trackpad swipe up.
It sounds like you're talking about fullscreen, not maximizing; it seems to just work that way. Unlike in Unity, though, the window no longer belongs to any desktop -- it's in its own space, all by itself. You can drag it between any of the other spaces, of course, if you want it over by the left again.


If you want to maximize your window instead of fullscreening it, ⌥ (option)-click the green(by default) title bar button. Note that in some apps, like Pages, this will expand the window to fit the content instead!