Most likely the popped capacitor is causing stability issues (output going up and down as you described).
It should be replaced with a part with comparable voltage, ripple current, ESR and life ratings. Failure to choose an appropriate cap will likely result in the cap failing prematurely.
A properly protected power supply should not be damaged from a blown output capacitor. I wouldn't recommend operating it until the cap is replaced.
It seems odd to me that the power supply is capable of self-damage. With 25V caps the max output ought to be below 20V.
I would use a sharp utility knife to cut the part from the glue, then desolder it and replace it. Make sure you note the polarity.
First off, I'm going to presume that the TV is out of warranty. If it is still covered, your best bet is to call the warranty line and get it fixed that way.
It's hard to tell from your popped resistor photo, but is there a heat sink to the left of it in the image that is black? There might be something under there that blew out too.
As for DIYing the repair, you could try swapping out that through hole resistor that blew. However, while it's common for low quality capacitor to give out, resistors don't usually pop unless there is an over current situation, so there might be a larger issue at hand here. It also looks like there are 0 ohm and a 47K (? hard to read) ohm surface mount resistors in the damaged area on the back of the PCB that might need replacing.
If it's currently broken, there's no harm in attempting a repair. Find a friend who does electronics to help you source the parts, especially if you think you'll need to replace those surface mount resistors.
As for getting a replacement board, rather than having it replicated from scratch, you should be able to get a replacement board. The various boards in TVs are usually modular (power, tuner, etc), and that appears to be the power supply board. Try searching for "TVMAKE TVMODEL Replacement Power Supply Board" and see what comes up. I know the Vizio 50" plasmas have a nasty habit of blowing power supplies and thus they're pretty easy to find replacements for online.
Good luck and stay away from your boyfriend's books lest they burst into flames. ;)
Best Answer
A lot of times you'll use glue/epoxy to protect against a mechanical shock (i.e. dropping it). Put it on the at-risk items, like the biggest capacitor, and you have more anchor points with less stress on the solder joints.