I have a Nikon D5200 and want to get an affordable off camera flash setup. I'm new to this, and have no experience or knowledge about flashes, so my question is: Would getting the MEIKE MK570-Ⅱ flash and Godox CT16 trigger/receiver set work, and would it be any good?
If you get that particular flash it already has a built in radio receiver that is compatible with the Meike MK-RC9 transceiver. You'd just need to buy a single MK-RC9 to place on your camera to fire the flash. However, you can buy a set of two MK-RC9 for about the same price as a single one. That would allow future expansion with another non-Meike flash in the future. Using the extra receiver instead of the flash's built in one just means an extra piece of gear and set of batteries to worry about and keep track of.
The Godox triggers linked in your question will fire the flash (or any other single pin manual flash with a trigger voltage below about 10 volts) you linked in your question, but they are Godox's old 433MHz technology. Godox has since moved on to a 2.4GHz system like most everyone else. 2.4GHz is superior to 433MHz in terms of range. In more complex setups using multiple flashes in multiple groups, 2.4GHz has even more advantages over 433MHz.
You'd probably be much better off going with something like the Godox TT600 manual flash that has a built-in receiver and transmitter for the Godox 2.4GHz system. You'd need a Godox X1T-N to trigger it (or XT-16/XT-32).
Yes, I just want to know if it would fire the flash. I don't really mind having to set power and zoom on the flash itself.
You might not mind now, but after about the tenth time in ten consecutive shots you probably would...
The nice thing about spending just a bit more on the Godox X1T system compared to using the Meike MK-750 II with the CT-16 system is that when you get tired of walking back and forth every time you want to change flash power, or you want to mix manual only and TTL flashes, you get tired of having three sets of batteries and their backups to recharge and keep track of, or you want to add bare bulb and/or studio flashes you won't have to start over with another trigger system - The X1T can handle all of the Godox flashes within their system. You can upgrade to manual or TTL flashes with a single rechargeable Lithium battery in each flash, add more powerful bare bulb or studio flashes, etc. without rendering your original manual only stuff useless.