Electronic – Can electronics be damaged by undervolting it

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. ;)

I know you said you don't want to run outlets everywhere, but I think a pile of wall warts is exactly what you want. Jameco has a large selection. 9 V sounds pretty good and that is one of the standard voltages, but you might want to consider 5 V (see below). Get something in the 500 mA to 1 A range.

These supplies are inherently isolated from the line, are usually short-circuit protected (check to make sure, you definitely want that), and draw so little AC power that you can string a bunch of outlet strips together without harm.

In the end, I think this will cost less and provide a better experience for the kids. I remember when I was a kid tinkering with this stuff how frustrating it was to have batteries run down, especially when you're not aware of when you are asking a lot from them. You also feel a lot less guilty abusing a power supply than running down consumable batteries.

You can start a fire with almost anything. If you do just the right thing, even a 9 V 500 mA supply can catch something on fire, but no more so than a 9 V battery and without the chance of the battery itself doing something bad and causing chemical burns.

If you are worried about LEDs getting damaged by them getting hooked up backwards, maybe you should get 5 V or 6 V supplies. Most LEDs can handle 5 V in reverse, and some 6 V. With 5 V supplies, you can eventually run logic cicuits directly without having to use a linear regulator. Lots of stuff will run well from 5 V. If you get 5 V supplies, get at least 1 A capability. That will be useful for running small motors. 5 W total power really isn't all that much.