I have not grown a Joshua tree myself, so the information I can offer has been gleaned from other sources. I understand that a Joshua tree develops a long taproot early after the seed sprouts. This is to enable it to access sufficient moisture in its native desert habitat. Although some sources recommended that Joshua trees be grown in deep, narrow containers, the reason given was to prepare the plant for transplanting outdoors in a desert environment. For a plant that will be grown only in containers, I think I would use a pot that is slightly shorter than it is wide. These are sometimes sold as azalea pots
or cactus pots
. The taproot will just wrap around and around the interior of the pot, which should not cause a problem.
One important point I noted was that the seedlings should be kept out of direct sunlight for the first year. I think your west-facing window is probably about right for the Joshua tree sprout, especially if it gets some shade from your bonsai plants.
You should probably repot the seedling fairly soon. In addition to the deep taproot, Joshua trees, like other desert plants, develop a wide, shallow root system. This enables them to absorb water quickly, as soon as it soaks through the surface of the soil. As a general rule, I try to keep my cacti in containers large enough to allow at least 2 inches from outer edge of the base of the plant to the rim of the pot. I would move a plant like your 'blade of grass' to a 3- or 4-inch pot.
As for watering, I use a combination of methods. I water my cacti and succulents from the bottom for about 3 waterings, and then for the 4th, I water from the top. When I water from the top, I really drench the plant, because my purpose is to wash out any excess salts that may have accumulated. In order to do that, I water from the top until water is freely running out the drainage holes, allow all the excess water to drain off, and then repeat the process. From late spring through the summer, when the plant is actively growing, I follow the drenching with some water in which I have mixed some liquid cactus fertilizer. The frequency of watering has to be determined by your specific plant. I think no more than once a month, or maybe even once every 6 weeks, should be sufficient during the winter months, but the plant will need more moisture during the spring and summer, when it is actively growing.
I use a commercial potting mix designed for cacti and succulents. You can make your own, if you prefer, but I've seen recipes online that call for quite a bit of peat moss, which I personally would not use for a cactus. Cacti grow in an environment where the soil is very sandy, with a high proportion of grit, and even small stones. It's usually neutral to slightly alkaline in pH, and drains very rapidly. (Peat moss tends to be acid, and retains water, which is why I avoid it for my cacti and succulents.) While it's fine to make your own potting mix, I recommend that you research the characteristics of the soil of the Mojave desert, since that's the only place Joshua trees grow naturally.
I hope that all helps you somewhat!