You'll need 14/3 from S1 to L1 to S2; 14/2 from a 15 amp circuit breaker to S1 or to L1; and 14/2 from L1 to L2 to L3 to L4. Connect all the black wires to the black 14/2 lead, all the white wires to the white 14/2 lead and all the grounds to the ground lead. Diagrams can be found in many Simplified Home Wiring books in your library or DIY store.

"We would like to keep the 3-way switches but also add another set of 3-way switches so we can control half of the recessed cans at a time. " You will have to break up the existing string of lights and add another switch at each location. Each switch will control half the lights. You need to add up the loads on the existing circuit to determine how much more it can support. You are generally limited to 80% of the circuit capacity (15 A or 20 A)

From the pictures of your wiring it appears you have everything you need present in the boxes. You first need to determine which common goes to the load, and which one is power. Whichever common is power goes to the master switch. With both switches removed there will only be one hot wire which is what we are looking for. If that is not the box you want the master in, you can still do it by re-configuring the wiring. The fact that your 3 ways are wired with 2 wire romex is not standard, and will make things a bit more confusing. One thing we need to verify is that both boxes are on the same circuit or we could have a violation, so check that out as well. Find that hot wire and we'll go from there. My guess is that it will be wire "G". This post was edited by joefixit2 on Sat, Dec 20, 14 at 23:52

"I admittedly did not try just isolating the box ground from the bundle to see if that was the culprit." Well, check it. I am going to assume you have a metal octagonal box with nonmetallic sheathed cable running into it. First, you have too many wires in the box if it is a standard box. Second, you might have two cables under one clamp in the box. If someone tightened the clamps too tight, especially if there are too many cables in them, and then someone forced too many wires into the box, you could have enough damage to a hot wire's insulation at the clamp to cause the problem. Make sure that the metal of the light fixture over the box is isolated from the metal junction box as well, or you would be shorting out through an alternate route.