What are the optimal conditions to maximize Peltier cooling performance and duration?
My peltier plate only stays cool for about a minute or so then begins to heat up, I'm not sure if this is supposed to happen or maybe I should lower the voltage I'm using?
  


This is normal behavior, your elements are not ideal, they have some resistance, and the amount of current they drain, the \$P=RI^2\$ becomes pretty high. Note that the equation says current squared. 

A solution to this is to add a heatsink and a fan. With just a heatsink it will still heat up to pretty high temperatures. A fan is definitely required. 


  
  What would the optimal circumstances be to maximize cooling performance? Is there an specific voltage or condition that would allow a for the highest yielding cooling effect?
  


A heatsink + a fan on the hot side will lower the warm side's temperature. This in turn will lower the cold side's temperature further. 

Some X voltage will give some Y temperature difference between the warm side and cold side. I'm not sure if it's linear or quadratic, or something else, but what I do know is this, higher voltage will increase some temperature difference, I just don't know by how much. Since you want to get very cold temperature, then you would need some higher voltage. I have no idea how much. This in turn will make the peltier heat up because of the non-ideal resistance inside of it as I mentioned earlier. So you will need an even larger heatsink and stronger fan. 

Look at a computer fan that is meant to dissipate about a 100 W CPU, it's not that big, it is however pretty noisy. Which is something you will have to live with, or get a 100 cm X 100 cm heatsink or some other silly size. 


  
  Lastly, I was wondering is there any way to orientate the conductors to have the hot and cold plates be on the same side?
  


You can get several peltiers and then have some upside down and some not upside down. This will give you hot and cold plates on the same side. I believe there are peltiers in the 1x1 cm² range. But I don't believe there's much you can do on your own with that peltier in front of you. Unless you desolder one entire side, desolder the ones you want to flip, solder back, solder back the entire side. This "sounds" easy but I would never attempt at doing it. If you wonder why you need to flip it, then it's because at the junction, aka where one part meets another part, there's two different metals. And you need to switch these two metals around. Either way, I recommend that you just buy smaller peltiers and flip them module by module. 

The ones you do flip will get really warm though. If you will get, say 30 °C temperature difference between the sides on one peltier and ambient temperature is 25 °C. And the temperature rise is 5 °C because of the peltier heating up due to non-ideal resistances. Then you will have 30 °C on the heatsink and 0 °C on the cold side, just barely able to freeze water. (I've done this in real life, was very fun). If you flip one of the sides you will get a little bit less than 30°C on the heatsink (because now one is cooling it) and 60 °C on the warm side. The total difference between these two sides will be roughly 60 °C. Depending on what you will be working on, that's a lot. 
But it's a good thing to know. 

Reversing the current on a peltier will make the hot side the cold side and cold side the hot side. No problem at all. 



TL;DR

Just get a heatsink and a fan. If you want to go even cooler, then get another fan and a larger heatsink and increase the voltage going to the peltier.