Electrical – Running 12V Wire Behind Walls for a LED Stair Light

Regarding your code question: no, you cannot hardwire the transformer.

I believe this falls under NEC 400.7:

400.7 Uses Permitted.

(A) Uses. Flexible cords and cables shall be used only for the following:

1. Pendants
2. Wiring of luminaires
3. Connection of portable luminaires, portable and mobile signs, or appliances
4. Elevator cables
5. Wiring of cranes and hoists
6. Connection of utilization equipment to facilitate frequent interchange
7. Prevention of the transmission of noise or vibration
8. Appliances where the fastening means and mechanical connections are specifically designed to permit ready removal for maintenance and repair, and the appliance is intended or identified for flexible cord connection
9. Connection of moving parts
10. Where specifically permitted elsewhere in this Code

At a stretch, if you could permanently attach the transformer, you could maybe consider this to be under 400.7(A)(8), but then you're still required to have a removable plug:

(B) Attachment Plugs. Where used as permitted in 400.7(A)(3), (A)(6), and (A)(8), each flexible cord shall be equipped with an attachment plug and shall be energized from a receptacle outlet or cord connector body.

Additionally, I think this could be interpreted as violating 400.8:

400.8 Uses Not Permitted. Unless specifically permitted in 400.7, flexible cords and cables shall not be used for the following:

1. As a substitute for the fixed wiring of a structure
2. Where run through holes in walls, structural ceilings, suspended ceilings, dropped ceilings, or floors
3. Where run through doorways, windows, or similar openings
4. Where attached to building surfaces Exception to (4): Flexible cord and cable shall be permitted to be attached to building surfaces in accordance with the provisions of 368.56(B)
5. Where concealed by walls, floors, or ceilings or located above suspended or dropped ceilings
6. Where installed in raceways, except as otherwise permitted in this Code
7. Where subject to physical damage

So I'd say, no, you cannot just hardwire the transformer in any way.

Additionally, you need to ensure the transformer (and all connections) are accessible for servicing. Transformers also get hot and need cooling - failure to have adequate cooling could cause the transformer to prematurely malfunction and/or lead to all the other problems of hot things in contact with flammable materials like your walls.

I'd find some area (such as under the stairs) that is accessible and unfinished, and run a new receptacle there. Then you can plug in the transformer, run the low voltage wires back to the LED strips, and you're good to go.

It's quite common to remotely mount transformers -- eg, it's very common for low-voltage under-cabinet lighting in kitchens that the transformer is located in the basement near the electrical panel or other unfinished (serviceable) area.

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.