Hotchoccy, you'll need to get some solid information about your soil situation before you can hope for success. If the sand is very fine textured, like flour or powder, you'll never be able to do much with it. Coarse textured sand has more possibilities, though that kind of soil is not at all physically stable....it shifts. You'll also need to find out about the water table and the sodium content of the soil. Do you know what the pH is? What about the quality of the water that will be used? I think that once you find the answers to those questions, you'll be in a much better place. You'll have to work with the sand because it's an almost impossible soil type to change. Mike is right. We can grow our plants in almost anything...as long as the fundamental needs of plants are provided in the right balance: water, air, essential elements....and sunlight. What are the typical nighttime temperatures? I'm curious.

I'm also growing meyer lemons in MN, so it's possible! 1) I'm supplementing with GE daylight CFLs during the winter, and that seems to be working. 2) Make sure your pot and soil promote good drainage. You don't want the soil to stay damp. 3) I'm having really good luck with a fertilizer called "Foliage Pro" that you can buy off of Amazon. The littlest bottle will last you well over a year. 1 teaspoon per gallon of water, and it doesn't hurt to add one teaspoon of white vinegar per gallon of water. I water about once per week or so. When you notice the surface of the soil is dry, then give it a really good dose of water, but don't water it every day. If your air is dry, you can try misting the leaves every once in a while. Good luck!

From reading your post, I assume that you are cultivating all of your citrus seedlings with the same care, and nutrition. Therefore, I doubt that the lemon's problem is due to a lack of nutrition. Lemons grow well in areas such as the California coastal areas, where the weather does not become too hot. Therefore commercially, lemons are not grown inland in the California desert areas. Metal Halide lamps generate a lot of heat. How, hot is the leaf surface temperature of your lemon, when under the MH light?

If I am not mistaken it looks like most if not all of your new growth is trifoliate, meaning its rootstock. (the top right shoot in the first picture may not be) Cut off all of the new growth that grows its leaves in 3's instead of 2's, its not your lemon. I personally have no luck growing my tree's indoors and would have put that tree out in April at your first posting and just brought in in when the temps dropped below 38ish degrees, I have left mine out to 34 with only minor leaf drop but mine may be on different root stock as well as they were in a more protected location from wind etc. I imagine you probably still have spidermites or the damage from them is hard to recover indoors and without proper sun, air flow, and fertilizer the tree is just going to stay in its condition. I am sure your window is great for wintertime but in spring/summer the best bet (in my limited experience) is putting it outside. My Meyer lemon looked pretty sad when I put it out in late march/early April but has since perked up a bit and my day temps were high 40's low 50's with low's in the 40's. Transition it out into a shaded area, then part shade, then full sun with 1-2 weeks between transitions, if your intense sunlight is too much a part shade environment or early sun and afternoon shade may be better but you'll have to experiment. At this point your tree appears to be dying and you have nothing to lose so I would put it out and start feeding it lightly with foliage pro or something similar, fertilizer spikes are notoriously bad options for citrus and organics don't usually work well in containers. I have a New Zealand lemonade tree that I accidentally let dry out and it dropped every leaf, now a month and a half later its got quite a bit of new growth. The magic of the outdoors is real for me, hopefully it helps you as well. It sucks to lose a tree you put so much time/energy/money into.