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.
Problem (mostly) solved. It turns out most dimmers have a way to adjust the minimum level of the dimmer. The way to adjust my dimmer was very unintuitive, but it allowed me to crank it down to about 10%, where it shipped with a minimum brightness of about 50%
Lutron has a nice video on their dimmer site that shows how to make the adjustment. Other brands of dimmers have different ways of adjusting the minimum output.
Best Answer
Some are. You can also get transformers designed to mount into a 1/2” knockout in a junction box. The transformer proper sits outside the box. You can also find transformers which sit on the outside of a metal box cover. These are very common in 24VAC.
These are typically true transformers. You may be using the word "transformer" loosely to mean any low voltage power supply. That may be a bit harder to find.
Any device you put on 120V power should be UL listed for the purpose, or inside a listed metal enclosure.
Another option, if there is a metal fixture involved, is to mount the power unit inside the fixture.
You do need to watch out for box fill limitations. Don't overstuff the box.