What's slowing down my Mac?

Your machine should not be this slow. If the below doesn't help, you can take it to an Apple store to run hardware tests. I've seen instances where faulty sensors, for example, would cause the computer to go into limp mode, and the hard drive was fine.

Have you tried resetting the PRAM?

There are some command line utilities that should be able to pinpoint what's using the disk (if that's the case). You can try iostat 1 - this will display general disk activity, refreshing every second.

To see more granular info such as the process, use "fs_fsusage": sudo fs_usage -f diskio

I'd first try to narrow down what's eating all the I/O using the above commands. You can always wipe and reinstall if you want, but if you restore your entire Time Machine backup, you'll be potentially re-introducing the problem. So I'd say only restore your data/ user account w/o the ~/Library folder if possible.

On the hardware side of things, these devices shipped with at least 7200 RPM drives. While much better than the slower 5400, modern OSes will still choke with excessive disk activity (spotlight, time machine, various system services).

Replacing the drive with an SSD on your own IS possible but you need tools, and have to be very careful handling the screen. So if you've never done it before I'd recommend paying someone to do.

EDIT: Also check your system log in Console... look for disk i/o errors

I know this is kind of a blunt comment, but after seeing your Mac specs, you really really should buy an SSD. If there is no other real problem (maybe there is) that is the single most obvious bottleneck I can see. ssds are fairly cheap now, so u can get a 250gb for well under 100$, and the hdd is easy to clone+ install, and your Mac will feel (and maybe be)5x as fast. If u still have warranty up u might have to pay a cert. Mac Tech tho.