It's always tough to diagnose from pictures, so this will be fairly general, but I'll try to touch all the points.
Any plant is going to follow the water, so a root coming out of the drainage holes doesn't necessarily indicate that the plant is root-bound. Bottom-watering encourages saucer-roots, and also is not that effective once a container is over 2-3" high. It's OK every once in a while, like if you're going on vacation, but not as a general habit.
If you're in a warm enough climate, these things thrive outdoors in bright shade, otherwise a cool sun porch is good. Even in a cold-winter climate, it might benefit from a summer outside under a tree.
I've heard many tales of plants which prefer to be pot-bound, but in 30 years of being a mad horticulturalist, I've yet to meet one.
Being impervious, plastic and glazed pottery perform about the same with regard to soil and watering, but if it's unglazed terracotta, a heavier, richer mix might be OK.
The thing about watering and pots and soil is not so much that we're watering a particular type of plant, it's that we're shuffling all of the components in order to allow us to maintain an aerated, evenly moist environment for the roots of a plant that fits in with our watering habits.
Top watering helps to freshen the soil, and a thorough soaking every once in a while ensures that dry patches aren't developing in the pot.
Never put any type of drainage-enhancing stones, pottery shards, gravel in the bottom of your pots. They all impede percolation and drainage, and reduce the amount of potting mix in the pot.
Taller pots are usually better than shorter pots, particularly for plants with sensitivity to excess moisture.
I can't really see the pot in your pictures, but I think it's safe to say it wouldn't hurt to give the plant a little more room.
I agree that some cuttings would be a good idea, and you could use the opportunity to shape the plant a little.
This is the time of year when these cactus need cooler temps and shorter days in order to set flower buds, so make sure that it's not getting any artificial light.
My idea of ideal conditions for any jungle cactus would be a mix that's mostly pumice (4 parts), with some peat-based potting mix (1part)(I like Sunshine #4), and maybe some Turface (1part). You want it to be fast-draining, with medium moisture and nutrient retention.
Christmas Cactus have aggressive root systems. They're really the kind of plant which would love to be sprayed with the hose every morning, and the water which fell on the soil would probably keep it moist. Of course, this is impractical indoors, so we build more moisture retention into the mix.
I've found that the best way to maintain indoor plants is to water them lightly and often, and once a month take them outside or to a sink and spray them off really well and run water through the soil. After that big drink they can go a while until the next watering. The actual frequency depends on the mix and the environment. You don't want the bottom of the pot to be constantly waterlogged, but you also don't want the top of the mix to totally dry out, since most mixes are difficult to re-wet.
Instead of trying to maintain the plant, try to think of this as maintaining the soil in a condition that would foster good growth. Christmas Cactus will show you they're drying out when the leaf segments get thin, but by the time they do that, some root tips will have dried out and died, and the soil may be very difficult to re-wet. Then, when you re-wet it, the dead roots will decompose, and could possibly cause problems. The thing is, overwatering and waterlogged soil can cause root die-off too, and when that happens, the segments will get thin, encouraging you to water more.
A lot of this is done by feel and intuition. Watch the plant, and when you see it starting to grow, feed it a little. Feeding should be balanced with the amount of light the plant receives. Indoor plants usually do pretty well with something like 1/4-1/2 strength Miracle Gro. If you feed and then see soft, weak growth, that's too much.
Don't buy "bloom" fertilizers. A plant is a complete growing system, and everything a plant needs to bloom is in a general-purpose fertilizer. Overdosing them with phosphorus is of dubious value, plus, certain soil types can develop phosphorus toxicity.
Having said all that, my grandmother grew a Christmas Cactus for many years next to a window inside her shed. It was in an 8" terracotta pot filled with garden loam, something no one would ever advise. It flowered every year, and had a large, sturdy canopy of leaves.
Sorry if this jumped around a bit, it's very difficult to edit these things on a phone. I just hope that some of this can be useful.
Also, it would enhance our ability to give advice if you edited your profile to include your USDA zone and some general geographical information.
