And when I say pain, I mean PAIN. I have flat arches and doing seated calf raises is excruciating on them. Does anyone have any solutions for my arches that won't slow down my workout? The trainer at the gym is saying I have to just put up with it, but it's such an extreme burn on the tendons it feels almost like they're tearing, though I know they're not. It sounds like I'm a wimp, I guess, but trust me, I'm no stranger to pain and muscle pains from working out I can deal with, I even like it to a degree, it tells me I'm accomplishing something, but grinding joint pains and tendon burns tells me I'm doing damage. How can I get past that? Does anyone have any suggestions?

Buy a set of Arch lifts for your shoes and invest in a good set of ankle braces and good workout/gym shoes.
Failing that, get your hands on some boxing wrap/tape and bind your feet and ankles yourself, adding a little padding to the edge of your feet to lessen some of the impact.
They are the only things The Pimp can suggest.
Best of luck in finding a solution.
Peace
The Pimp NeonBlack

I wish I had seen this sooner. 

The joint at the ankle is a 2nd class lever and is designed for force.  My first inclination is to check your form.  Like any movement, it should be controlled and smooth.

However, one recommendation I can make that may help aleviate the pain is take a water bottle and leave it in the freezer until frozen.  Take it out and roll your feet over it.