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.
Maybe the connection to the switch is broken and it is returning to what it thinks is the previously set dimming level?
Or maybe the overheating or short-circuit protection in the LED-driver is activating and turning off the outputs.
The correct wiring is as shown below
Note that the "switch-dim" connection should be connected to a momentary pushbutton switch - not to an external dimmer and not to a normal two-position lightswitch.
Best Answer
I found a bit more information and thought I would update this. The code has changed and I believe specifically section 8-304 of the most recent code has been amended for situations like this. The limit is 12, unless you are installing devices that have a known load. In the case of a LED light which is hardwired in, or a smoke alarm, you can put as many of these devices on the circuit providing you don't exceed the 80% load. This applies as long as you have no points on the circuit (such as receptacles) which would be an unknown load (since you don't know what someone is going to plugin). The moment you add a receptacle the limit drops back to 12. Hope this helps!