UPDATE!
I replaced the .33uF cap with a .27, as specified. I also put pin #6 on CA3162 to +5V, so make the chip run in the 96Hz mode. These things made no real difference, other than it updates very quickly now.
I replaced the battery. I had not noticed that it made it down to 4.5v, and that the 7805 regulator was cutting it to 3.2v. That was stupid on my part. Then after hours of adjusting, I realised that the number one digit was not coming on sometimes. I ended up soldering two of my spare meter leads to the board to constantly monitor the LM35, so I could compare it to the screen. After another hour, after checking all my connections again, I pushed the LCD display all the way into the socket. Immediately all three digits lit up, and I could now see that this thermometer reads up to 99 degrees C, to the tenth of a degree. I lit up the number one digit decimal point by hooking it up to ground, and it magically made sense. Now it reads exactly what the LM35 puts out, .214vdc on pin 11 on CA3162, and the display reads 21.4, room temperature.
In summation, it is working perfectly, and much more brightly. I think that I spent over 40 hours on this project, and I am happy that it now works to my satisfaction. This could have been avoided if I had just pushed the LCD all the way in to begin with. I did't want to force it at first, and later, I fell into the trap of not double checking it.
I already used Solidworks and the laser cutter at school to make a plexiglass base for it to sit on, and tomorrow I am vacuum holding a cover for it. I will update again with pictures.
Thank you all for the suggestions, I investigated every single one.
Cheers! Josh