Ju00Ls

Great project!   I'm really enjoying following along with your progress.    

Just some comments from an artist (not an electrical engineer);   the metal box to me looks very scary if it's not grounded.  My kiln draws ~22 amps and I made sure the exposed metal sides of the kiln has a good ground path to earth that will easily dump enough current to allow the main breaker to go. (checkout ground bond testing)
It looks like the AC power runs very close to the TC cable?   to reduce noise you could try and keep the AC line away from the TC cable and also using a shielded TC cable.   The TC anomalies (open / short) could be linked to the i2c  library?  the library not accounting for clock stretching maybe?    as a workaround you could discard the values and perform another couple of temperature readings after a short delay and only branch your logic if multiple bad readings are present.
When i built my own controller I too was getting noise;  what I did was i took a few TC readings and discarded the upper and lower values,  then I did a check to see if the readings were stable within a +- 1c value;  if not,  I output a buzzer tick noise;  this greatly improved my temperature readings,  and what I noticed was on odd occasions and when my hot water boiler came on the ignition caused a lot of noise on the TC wires,  the noise was enough to throw a reading wildly off, but with the software error correction logic it simply just waits a couple of powerline cycles and takes another series of samples,  this for me works really well. 
When I read up about PID for my own controller I didn't really understand it, so i just rolled my own code;  I can't remember exactly how it works without checking the code but 
the thermal hysteresis for me was only an issue at the kilns lower temperatures,  after the kiln got hot it wasn't an issue, so my code logic just switched to a different mode when the kiln hits a certain temperature.
 
Not sure if that was helpful lol but keep up the great work!

Thank you. I've never used one of those, so I wasn't sure how it was powered.
To clean up the box:
1. Replace the power cord with a 6ga 3 wire cord. You can reuse the plug, just get a new cord. I buy the SEOW cords from McMaster, about $35 for a 10ft length. They're super flexible and easy to work with. Use a cord grip, or just a simple clamping cable connector to hold the cord securely in the hole in the box. Use a snap-in plastic grommet to hold the TC wire, and another for the controller power supply. You'll have to figure out which sizes you need for each cable. They not only protect the cord from the sharp edge of the hole, but keep the cord from pulling out if you accidentally snag it.
2. You can go with the single leg SSR if you add in a mechanical main relay. Every kiln system I've seen uses a definite purpose contactor (DPST-NO) for the main. Because it is big and your controller has low voltage output, you'll need an intermediate relay (pilot relay) between the controller and contactor. So for the pilot relay you'll need something that has a coil that will work with your controller, either single or double pole. L&L uses their standard 25 amp 2 pole 12VDC coil relays for the pilot. In the Skutt KM-1 external controller, they use a small single pole relay and just run the power for the other side of the coil directly off the main power. I've attached a wiring diagram of the KM-1 which should give you a good idea of how you can do yours. Either way works. The contactor, then,  will need a 240 volt coil. You'll have to set up your programming to turn the main relay on and keep it on when the kiln is firing, and off if there are any error situations, or at the end of the firing. The most important error would be to shut it down if the temp is climbing too quickly, which would indicate a stuck relay or TC issue.
3. Make sure you ground the box.
4. If this was my project, I would use a larger box so it's easy to work in, put in a terminal strip where the main power comes in, and distribute everything as needed from there. It'll make the wiring much simpler to deal with, and easier to see what's going on. Terminal connections are much easier and cleaner than wire nuts.
I think we're all on the same page here. You've got the thing working, which was the hard part (congrats!), now it just needs cleanup and safety concerns addressed, which is mostly just busy work. It's a great project.
@liambesaw I had a small test kiln last year that got really hot during firings. Because the kiln and box were only about 10 inches tall, there wasn't enough draft created in the box to keep it cool like it would in a taller kiln. The controller was getting very close to its max operating temp. So I installed a little computer fan in the bottom of the box, and it made a huge difference. I agree with @Bill Kielb that it would be good to avoid adding in more parts, but it will definitely work if it comes to that.
I've also attached a pic of a Skutt with SSRs. You can see the big cooling fins on the box.

I got tired with going down to the basement to make sure my kiln was working properly and running on schedule so I started (forked) a project to control my KS-1018 using a raspberry pi. With this I can:
monitor my kiln from anywhere using any device that has a web browser (phone/tablet/computer) easily create new schedules, edit existing schedules, including infinite ramps & soaks accurate PID control get schedule cost estimates and actuals If anyone is interested, here is the github link... https://github.com/jbruce12000/kiln-controller
The cost for everything I bought was under $200.00, but my time investment has been high.  Again, this is for geeks that were born with a soldering iron in one hand.