You can use any NM or even AC90 or THHN/THWN wire that is 14 ga or thicker. The 15A breaker demands a minimum of 14 ga but there is nothing wrong code wise with going with a thicker wire except for convenience of working with it and the price. In the future if someone saw 12 ga wire they might assume a 20A breaker. It's not best practice to mix wire sizes on a cc't, but it is code compliant.

Some other points to consider, the box fill calculations are different for #12 and #14. Fewer #12's allowed in the same box. You are probable using 15 amp switches on the #12-20 amp circuit, then your circuit would need to be protected at 15 amps and you ran 20 amp wire. Every time the AC cycles your lights will blink or dime breifly. In a 3 way switch you need to keep track of the common and traveler wires. How yo gonna do that with 2 seperate wires. Then when you get to the light box how you gonna know which white is neutral or traveler or common? and which black goes to the light? Not good practice. Generaly the black and white of the 3 wire between the switchs is the travelers and they go to the same colored screws on the switchs. The odd colored screw of 1 switch will be fed hot and the odd colored screw of the second switch will be the common going to the light. If you need a wire diagram I would be happy to fax you a couple.

What ever you do, do not us irrigation wire nuts. Don't know where you are located but if you are near a large city or in one there should be a Ewing Irrigation Store close. Ewing1.com. They sell 3-M DBRY splice kits meant for higher amps than irrigation connectors. You have less than 50 watts on your 12/2 wire now and you are talking about adding another 15 watts for a total of less than 70 watts... ... ... not an issue what so ever unless you are planning on running those two lights over to your neighbors yard or you live on 10 or more acres and want to put them at the top of your driveway. Is your transformer multi-tap? If so put the 12/2 on the 15 volt tap that way you will have enough oomph to drive the last LED. The industry standard for 12 volt LEDs is an operating range of 9 to 15 volts. Less than 9 volts and the LED will not come on, more than 15 volts and you will burn out the circuit that drives the LED. As always take any advise with a grain of salt and read the instructions that came with the light fixtures and through the salt over your left shoulder. Hope that helps.

The green wire is the electrical ground for the new fixture. The bare ground wire is the ground for the metal box (required by code for metal boxes). You just connect them both together with the bare ground from your 120 volt wiring. However, for your situation I'm not really sure you have 120 volt wiring and a standard box. All I see in the picture is lamp cord. What exactly are we seeing in the picture? Is it a new fixture housing just laying on a counter or is it mounted on something? Do you have a picture of your NM house wiring? Bruce