If the noise bothers you, build a box about 3x as large as the compressor with air holes covered in filter. Line the outside of the box with clay and secure the compressor to the box. The high density of the clay will absorb the vibrations of the sound, so where ever the holes are, the sound will escape (so better face them to the floor). The extra size should allow for air movement and heat dispersal. I have never done this for an airbrush compressor, but I have seen similar things done for industrial compressors.

You can make the box out of just about anything, and the clay can be anything as long as it is dense (you can get it pretty cheap just about anywhere). Any dense material will work, the more dense the better.

As for CO2, I have used smaller tanks in the past, and they work pretty well early on, then loose pressure quickly. You might want to check out other forms of propellant (such as argon gas) which could be cheaper to rent. Argon can be dangerous, now that I think about it. It's not a poisonous gas (in fact, it is completely non-reactive), but it displaces oxygen, which can lead to suffocation. So use your arc-welder in a well ventilated area (which is what argon gas is usually used for).