First off, I'd suggest a pair of gloves with some duct tape wrapped around them to save your hands. No need to pick up a ton of glochids :) I can't much say what's causing the red spots and leaf drop, but if I had to guess, maybe rot? How well draining is the soil? If the spots are soft and the soil stays we for long periods of time, that could be what it is. If the spots are soft, you should probably cut out the soft parts, and if the soil stays wet for long periods of time, a soil transplant is probably in order. This is simply pure speculation, but another possible reason as to why it's loosing branches might be because in their natural environment, this type of cactus gets large and tree-like, and maybe it's simply naturally shedding its lower limbs. But since it looks like you've got a Brasiliopuntia, I wouldn't get overly upset about anything falling off it. If all else fails, you can root pretty much any part of it. The leaves are actually pads that can be rooted to start new plants, and you can root the branches that are falling off too. I really don't know if you could re-root the main trunk. With such a big specimen (an a nice one at that!), I think you'd have to be awfully unlucky to loose it permanently. To root a piece, just let the cut part dry for a day or two, put it in some well draining soil, and wait until you see new growth. Then give it a bit of moisture every so often. Opuntia family cacti are fairly easy too root, and it looks like you're not in short supply of starts if you mess one up. As for soil, cacti are pretty good about growing in low nutrient soil. As long as the soil dries out fairly quickly, you should be fine, although I don't think a nice dose of fertilizer during the growing season will hurt it. Personally, I mix a good bit of perlite (pumice works better, but I've never gotten any) in with my cactus soil to improve drainage. I can't quite tell what you're using from the picture, but it might not like ordinary potting mix. As for light, it probably likes a lot. You might want to consider moving it outdoors during the summer. In the winter, I guess there's not much you can do, but it shouldn't grow too much then anyway. The grow light might help, though. As for the draft, Brasiliopuntia are a more tropical type of cactus, and I'm not too familiar with tropical cacti, but I don't think the chill would be that much of a problem. If its survived it in the past, I wouldn't worry about it now. If this is its first year in the cold, I'm not sure what its minimum temperature requirement is, but as long as its indoors its probably nothing to worry about. For water, I would personally water it more during the summer, but that would probably make it grow faster, so as long as its doesn't mind once every three weeks in the summer, it should be fine. I think once a month in the winter should do, although, again, since its more tropical, it might like more water, but as I said, I've never had one of these. I know that lack of water can cause Christmas cacti to loose pads, its possible that this works the same way. (Yeah, I know, maybe too much maybe not enough isn't very helpful) Good luck, and you're probably better off getting a second opinion.

There is a small chance that it is natural ageing but it is more likely rot, I think. Is it spreading? If the base rots it usually becomes soft and the plant falls over, but with some fungal disorders the plant remains upright. If it is rotting, it must be cut at approximately half height (the tissue must not be discoloured, it must be cut higher up than the brown area on the skin), dried and rooted as shown in the link. I would wait for several months before potting it (this would be a large wound), but that seems to depend on the climate. The shape of the top shoot is not normal. It indicates that the growing point of the main stem is damaged. That is not too bad, it is probably easier to cut off and root than the major surgery needed for the main stem. It is also thinner than it should be, which could be related to root damage. It does not look like typical etioliation, even if I suspect that it would hav looked bette if it had been grown in stronger light.

Jocelyn, that's why I asked about soaking. What you did is exactly what I have done for my TCs. Then one day someone described how they soaked their plant and I realized that I wasn't doing what was suggested. Do try to removed most all of the soil from the roots of your plant before repotting. If it is hard, like asked above, it will still interfere with getting water to the roots. Be patient and soak and swish with warm/tepid water. Sometimes a bamboo skewer will help break up the hard soil without ripping off all the roots. Your plant will thank you. Good luck! Come back with any more questions and do let us see your plant after you finish. We love a success story!

As garrett said above, it's starving for sunlight. Without sunlight there's no photosynthesis. Without photosynthesis, the plant starves to death. Cacti, especially Pencil Cacti need full, direct sunlight. Place it outside.