The original Apple hard drive is most likely the original source of the system slowness....the hard drive is either worn out or failing, both can cause performance issues. An SSD upgrade would have been the better option to improve performance since it can greatly increase performance. Memory upgrades rarely provide any performance improvements....it is very easy to use Activity Monitor's memory tab to judge whether memory is the bottleneck.
On the 13" (mid-2012) model, the internal hard drive cable tends to have a high rate of failure....especially when using an SSD, but can also have problems with a hard drive as well.
You can check the health of the hard drive by running DriveDx (free trial period). For a hard drive, usually any "Warning" or "Failing" notices indicates a worn out or failing drive respectively. While supporting thousands of our organizations Macs, I have discovered that a hard drive should be replaced for either condition. Unfortunately drive health monitoring apps are not quite so straight forward with SSDs....these apps can alert a user to a potential issue with an SSD, but a manual examination & interpretation of the health information for an SSD is necessary to determine if it is a real problem or whether just normal behavior of the SSD which can be ignored for now. Feel free to post the complete DriveDx text report here using the "Additional Text" icon which looks like a piece of paper.
As @Servant of Cats has already mentioned, Macs are very picky about the memory they use, so it is best to only use memory from Crucial or OWC. Even then, it is important to only use the exact part numbers identified as compatible using the tools on their respective websites. For Crucial, any legitimate reseller will specifically mention Mac compatibility for the modules which are compatible with a Mac. In fact, the plastic container holding the memory should say either "For Mac" or "Mac Compatible" on it (at least they did when I last purchased memory for a Mac several years ago. You can no longer buy RAM based solely on technical specifications since RAM has a lot of technical specs which are not fully listed by the memory vendor and definitely not provided in Apple's technical documentation. I have not purchased memory based solely on technical specifications since 2005 (for the PPC Macs....it did not work out well for the Intel Macs).
Personally I would forget about the memory upgrade and instead invest in an SSD since that will give you the best performance boost & value. The Crucial MX500 SSD is a good choice as it is usually well priced, but avoid the Crucial BX500 series as it is a low end budget economy model with lots of issues. In fact, most consumer SSDs today are just low level budget economy models.....it can be very difficult to tell them apart anymore. Plus you can no longer trust any SSD reviews because many of the SSD manufacturers are now changing the internal SSD components without listing any model number change...some may build them with two different designs, so it is like the lottery whether you get a good one or a bad one. Another choice for an SSD is an OWC Mercury Electra/Extreme 6G SSD. I don't personally care for this OWC SSD (older version was Ok), but OWC generally makes good products and has excellent tech support (some of it is just my personal preference for regarding things which probably won't affect most people).
