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.
No matter what you do, yes, you have to follow electrical codes (in North America, the NEC), and this means proper enclosures for any mains voltage stuff. The NEC doesn't have anything to say about low voltage (<= 30V) wiring.
Essentially this comes down to preference, there's no right answer as to what is 'better'. A hard-wired solution can have a neater final appearance, but it's also more work to install and very difficult to remove.
Since what you're doing is kind of non-standard, I'd personally go with something easier to remove.
Install an outlet somewhere up high and near the door frame, then install a light switch (wherever is convenient) wiring it so the switch controls half the outlet. I'd look for a plug-in type transformer ("wall wart" style), wire it to the LED strip and plug it into the controlled outlet.
If you trim excess wire from the transformer (and maybe even run a small bit of stick-on conduit) it will still be very neat, and the benefit is you can completely remove the LED strip and be left with a standard plug and switch. Later someone could plug in their own light to this, or even convert the outlet to an actual light fixture (depending on where you put the outlet).
Best Answer
You do not need to
At that low of a current draw voltage loss will be negligible. The big concern about voltage loss with outdoor lighting was important with incandescent and halogen bulbs, not LED.
The calculator I used would put end-of-line voltage at 11.59 in your setup. Your LED bulbs will operate correctly down to minimum voltage required to power the fixture. Typically anything higher than 8.5 volts is acceptable; check your specific fixture.