What watt power transformer should I be using?
You can use a power supply like the one pictured, rated for the amount of LED's you will be driving. Read the specs on the lighting. They should tell you how many watts per a given length of the strip. So if it says, for example, that needs about 1 watt per meter, then a 15-foot section would require a 5 watt power supply. Take your total length of 76 feet, and divide by the strip's requirements. It wouldn't surprise me if you ended up somewhere in the ballpark of 20 to 30 watts.
Can I run 14 gauge wire directly from a circuit 15 amp breaker to the transformer?
No. You can't put the power supply inside the wall, and you can't bring the wire out of the wall without going through a junction box. The best way to do this is to put a cord on that power supply, and plug it into an existing outlet. There is no need whatsoever to add a new circuit or connect these directly to a circuit breaker.
Can I plug in all 5 circuits of LED strips on top of each other to the output of the transformer?
Yes, you can wire multiple strings in parallel at the power supply. If the sections are short, you can also wire them end-to-end. Double-check the specs for maximum length of a single run.
Is this the recommended way of doing what I'm trying to accomplish?
The recommended way is to plug into an existing outlet. If your lighting requirements added up to 1000 watts, you'd want a new circuit. At 20, 30 or 50 watts, it is ridiculously small in household circuit requirements.
I would also like to add a on/off switch, do they make a wall switch for 12v?
Any switch can be used for 12v. I would put the switch before the power supply. Put a cord on the power supply like this cord with switch.
Can low volt wires come right out of the wall, or do they need a junction box before being connected to the LED strip lights?
The low voltage wires don't need a junction box, but they do need a junction plate, like those used for phone jacks.
It sounds like you are right at the current limit for your power strips and your remote control switches. If you look at the plugs of the lights, you should see a rating that says how many Amps it uses. Add all of those up, and make sure you're not exceeding 80% of the rating on your power strip or your remote control switch. So if your power strip says 15 Amps, then make sure your lights don't add up to more than 12 Amps.
Best Answer
An arc discharge light, like a fluorescent or metal halide, is a dead short that happens to make light. (It is an arc, after all). They would flow infinite current if you let them, which would burn then up fast. You stop that by putting a current-limiting ballast on them. That's what a ballast is, it's a current regulator.
LEDs aren't quite that bad. But they are current-hungry, and must be current limited. If you are driving a raw LED, a small increase in voltage will cause a big increase in current. Their voltage-current curve is steep.
The best LED-based appliances use an active driver circuit which actively limits current. These active drivers are "magical"; they can input a wide band of voltages from 80 to 306 volts AC and outlut exactly the current the LEDs want.
Low-voltage consumer gadgets like Christmas lights often use resistors as current limiters. Resistors are dumb, and not magical at all. They don't regulate current, they just make the voltage-current curve somewhat less steep, which is "good enough" for working with one expected voltage. They also overspec the LEDs somewhat.
You raised the voltage to 125% of normal.
So you can see your "little voltage bump" is really putting the LEDs on the griller. You are leaning heavily on whatever overdesign was built into the LED spec.
The bigger question is not why this failed, but why the other one is still working! You seem to have made yourself into a "QA department" for these lights and are discovering the ones which were not overbuilt.