I run no more than two cables, 12-2 or 14-2, through a 3/4" hole. Your 12-3 should be by itself passing through the stud, as another cable could cause binding and abrasion damage to the cable sheath. Multiple 3/4" holes through the center of the stud will not appreciably weaken it, as long as they're not close to a large knot or other cut, notch, split, etc. The common neutral configuration will not give you problems as long as the GFCI is downstream of where the neutral splits to two circuits. Bringing the wire through the drywall and cabinet frame into the fixture would be my choice, as long as the fixture is designed for romex connections.

I just put in the Maxlite from EL at 18/12 in and I love them! You may find one 12 in bar per 30 inches leaves definite bright and dark spots, especially with the 33/12 in since it will be so bright to start. I'm not a lighting expert, but I think you would have better coverage with 2 x 18/12 in per 30 in cabinet. If your bottoms are flat, you can run a continuous strip all the way down. It's a nice product - small, diffuser included, and nice light. Easy to hardwired too, I just used shallow metal junction boxes under the cabinet for each section (each seperate area of cabinets, not each cabinet). If I were you, I'd call EL and speak to one of their engineers. Normally you can send your plan in through the contact feature and get a quite in a day or two, but I bet if you speak with them first you could probably get a quote that day. The advantage of having them design a system and give a quote is they will be able to set you up a shopping cart with all the parts you need already in place. You just log on and check out. If you use EL, sign up for their newsletter first. They will send a coupon for 10% off your order. I have nothing but good things to say about them. I called a few times with questions, and they were great. I'm very pleased with the lights.

Undercabinet LED lighting is a great DIY project... Here's what I did. Works out to about $5/ft. Links below are clickable. Here's a video below - this guy used connectors - I thought soldering was easier and cheaper. Installing undercabinet LED strips I suggest watching other videos also, search youtube on "led under cabinet lighting installation" and you will find several. I installed about 2-1/2" from the face of the cabinet to avoid shadows. You'll need 120V for the power supply. Lots of different type LEDs - brightness, temperature (color). Here's a list of the things I ordered from Amazon: power supply - $9 LED strip lights - $15 for 16 feet 26 gauge wire - $7.60 for a spool to connect the LED strips (solder) Aluminum channel diffusers - $27 for 5 pieces of 1 yard each Dimmers - $10 for 3. Optional parts. I never seem to use mine. Since then I've added LEDs to my RV, a closet or two, and plans for more.

Probably. However, I don't know much about code requirements, if any, for low voltage wiring. The OP says, "For each puck light there is a normal electrical outlet in the cabinet above. No wires or cord are visible inside the cabinets." That would suggest that the receptacle is being used as a junction box. Now that I re-read the post, when he said "about 1 in. of cord is visible between the puck light and where goes in to the wall," maybe by "cord" he actually meant NM cable. That would be OK, assuming the cable isn't exposed to potential damage. It does strike me as odd for the installer to put receptacles inside the cabinets, though. Has anyone else seen that done? Another thing seems odd about this post. If these are LED lights, how could the OP change the bulb?