Check the version compatibility for any theme before you try installing them - particularly with an older Moodle.
You say:
(The organization probably doesn't have the budget to upgrade, in case you're wondering)
What budget would you need? Done properly, it shouldn't take too long to do and can be done with a little overtime for someone, or time off as a straight swap. You do this Friday night after work and come in Monday lunch time. Practice, practice, practice. Do it often enough and an upgrade can be done and dusted in minutes. Of course, testing might take a little longer, so depending on the size of the Moodle, that is where the time can be consumed. Do it on your test server, and if you don't have one, build one, any 5-7 year old desktop is likely to be useful for this, if you don't want it to be connected to any other part of your network. This is how I started, on a budget that would be described as aspiring to be a shoestring.
Develop an upgrade policy, one that's consistent and sustainable, say once per year, at the time of least use is your best bet.
Code customizations are different, because you will need to ensure those are compatible before you upgrade your production site. But is there any reason these need to be maintained, can an existing, third party plugin be useful? Can the new core code do the things you need?
You unzipped your new theme and added it to the themes directory on your server via ftp. Why? You can use the native file manager of your server to add a theme, you don't need to ftp it if you have full admin access to the server. Don't delete any theme you have used until you have changed to to something else in Moodle, I wouldn't delete any theme that is core, they can be required by any new, third party theme. Do not delete anything manually in moodledata, that's asking for trouble, stay right out of it.
The HTTP 500 error is a generic error that describes not a lot at all. Have a look at your server logs, that might give you some idea of what the server was actually doing when it threw the error.
You refer a couple of times to "admin plugin" but exactly to what are you referring? The Plugins Manager? Themes manager? Did you go through the Themes manager to try installing your Theme Simple? You sort of indicate you have seen it, but have you selected it as your preferred theme?