"Remove" is only there if the page doesn't specify uninstall: false, e.g.,:
page(name: "myPage", title: "My Page Title", install: true, uninstall: false) {
// ...
So, you can control that much. You will likely want at least one page that has uninstall: true, but (presumably) to avoid users accidentally removing complex apps and perhaps lots of child apps and devices, some apps hide this button except on a special page you have to click through to get an "are you sure?" kind of thing.
The "Done" button will appear if you have install: true (and possibly even if you don't; I don't remember what the default is, but I always specify). If you have a nextPage: specified, you'll see a "Next" button instead of "Done" (adding to the list of things I don't remember, I don't reacall how these two interact, but that would be easy to test until you find what you want.
Finally, it's kludgey, but if you want to hide "Next" (e.g., until some inputs are satisfied without making them actually required), you can do so with a bit of JavaScript trickery:
(Subject to change at any time, I guess, but used internally so it's not totally wacky.)
So, you can't trap "Remove" or "Done," but you do have control over if or when they appear (and even some control over "Next," though I probably wouldn't do that unless you really need to).
Yes, you can use a input type: "button" // ... to place a button in your UI. This is, presumably, how RM does the same. When pressed, it will call a method that must exist in your code call appButtonHandler() with a single String parameter that matches the button name, so you can do an if or switch inside to act accordingly. If you add submitOnChange: true to the input, the page will also refresh (though you may want to ask why you're doing this, as maybe it would be better to just do submitOnChange with the other inputs in the first place).
Probably? But unless you need it in logs, I'd just look at the "App Status" page (gear icon in the top right) to find the information you're looking for--it's all there, too.