Lighting – Why do the LED strip lights only work if I reverse the connection polarity

I'm not familiar what that product specifically, but the "amplifier" in this type of lighting setup generally only reads the input from the previous upstream light strip and provides power from its power source. Therefore, the amplifier draws no current from the upstream lights.

Instead of 2 power supplies, you could use one power supply larger enough to satisfy all of your power needs.

To answer your questions specifically:

1. Yes.
2. No. Power supplies are rated for the current that they CAN deliver. Using a power supply larger than needed is fine.
3. Again, not knowing these devices specifically, I am going to guess no based on my experience with other, similar products.
4. Generally no, provided you do not draw more power than the supply is rated for.

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.