Where is the ability to install custom GreaseMonkey scripts? I've installed greasemonkey. My dialog box under tools only has GreaseMonkey -> [Enabled, Manage User Scripts..., New User Scripts..., User Script Commands (greyed out)]
Ok, the right answer here is to rename the file to a .user.js (literal) extension - which userscripts will do for you - and link to it in a webpage. There is no longer an Install Script dialog or anything of that sort that permits you to browse to a .js file and simply install. (that I know of)
Google Chrome on Android so far doesn't support plugins, of which Tampermonkey is necessary to run userscripts, so here I recommend Firefox for Android. You can install Tampermonkey addon in Firefox and then start with your favorite scripts. Note that all userscripts are not compatible with Tampermonkey such as SE-AutoReviewComments for which you need USI- Unified Script Injector addon.
Stack Apps is a Q&A site for apps, scripts, and development with the Stack Exchange API. People discuss about developing around SE, and publish their projects there. You can find a lot of userscripts in the scripts section.
When you encounter some issues with userscripts, try removing and reinstalling them, as well as clearing/resetting your Firefox addon, and see if the issue resolves by itself. Usually this solves 99% of them.
Tampermonkey or any other userscript manager can no longer be added to Firefox as of writing this answer. It was never even possible with Chrome. However, You can instead install actively maintained android firefox fork iceraven that supports extensions like tampermonkey, greasemonkey etc.