This model has been out a while, so disassembly instructions should be up on iFixit. You'll need to decide whether you're going to stick it out with Apple, or whether you're willing to sacrifice the warranty and start replacing parts yourself.
The least invasive thing I can think to try would be to install onto an external drive, boot and use that for a few days to see whether the issue is independent of the internal drive and cable, but this a diagnostic, not a solution. I'm not sure what to even have apple replace anymore. I run a business off of this machine, as well as doing graphic design and publishing work for a non-profit student organization. Her computer, opens it immediately, and has no lag working in the exact same files that I'm working in. For example - when I'm working in Indesign, if I try to open it, or even if I click the "Type" tool, I get a beach ball for 5 minutes.
My wife runs the same applications I do (Photoshop CC, Indesign CC, Safari/Chrome, MS Office, etc.) and she uses a Late 2013 (purchased new in July 2015) 21.5" 2.5GHz i5, 8GB, Iris Pro - and everything opens up quickly. Even a simple task such as opening About this Mac can take 5 minutes. I've checked disk & file permissions, and end up with no errors. No HDD bad sectors, no RAM errors, no CPU errors. I've run hardware diagnostics multiple times, and nothing fails. My system takes 20 minutes to start up, opening apps takes what seems like ages, and I'm back where I was in December. I've reinstalled OS X twice now, but this time, the first two days still haven't provided me any relief. It stayed that way for most of January, and then started acting up again in the beginning of February. I started the computer up when I got home, and it 'seemingly' worked perfectly. Logic board, bluetooth/wifi card and Hard Drive.
He also noticed that my bluetooth keyboard and trackpad kept disconnecting at random times. The genius there noticed that even after the initial startup, loading apps like Safari, or System Preferences took about 5-10 minutes each. I ended up getting frustrated so I brought it into the Apple Store (I have applecare on this machine). Just a black screen with an Apple logo which would sit there. Anyway, boot times would take an hour plus some days, and sometimes the system wouldn't even boot at all. I manually back up my files to my backup server, and then restore my files manually, making sure I do not carry over any of my preference files.
I should note, that I never restore from Time Machine backups. I was reinstalling OS X every week (once every two months prior), as the first two days after the install were the only days that it would run correctly. In December 2015, my system was unusable. Because of this, I stopped using VM's (I was also given a work laptop so I could stop using my iMac). I could click the Safari icon, and then get lunch before it would be open. The host OS during this boot time, is completely unusable. If I even open a virtual machine now, what took 15-30 seconds to boot now takes upwards to 20-25 minutes. Once I installed Yosemite, everything went downhill.
I had no speed issues, I was able to use both the VM and the host OS. Prior to installing OS 10.10, VM's worked incredibly well.
I work(ed) a lot with Virtual Machines, as I build and test software on different platforms.
Side question - I know the serial number and MAC addresses of the WLAN/Bluetooth & Logic Board's can be changed using Apple's utilities, but I thought that disk drives weren't? I have never seen a utility to replace a drive serial number (and I replace disk drives in server nodes and raid arrays on a daily basis). that's what I was told, at least - I question it because my HDD's serial number never changed, but it was explicitly written on the work order that the drive was changed). I have brought it into an Apple Store, who has replaced the logic board, the bluetooth/wifi card, and the hard drive (well. I purchased it in January of 2014, and ever since day 1, it's been a nightmare. I have been experiencing so many issues with my Late 2013 iMac (14,3 - 21.5" 2.9GHz, 8GB DDR3, 1TB HDD, Nvidia Graphics).