Planting depth is unimportant. Cacti that root from a cutting understand where they are in the soil and will adapt. They sense where the soil is and send
roots out into it, and they grow up from whatever side is oriented to the top. If you lay it flat, roots will grow out the bottom and new segments up at the top.
How much you water it will determine how fast it grows. Cacti have adapted to growing when water is available and going into stasis when it isn't. Which means you really have to let the soil dry out bone dry between waterings. If you give it a good soaking once every two or three weeks, that would be a good watering schedule to get maximum growth. I did that with my aloe plants when I lived in Las Vegas, and after two years they were HUGE.
If cacti go into a prolonged, months long drought, they can shrivel as they slowly lose stored water, but after a soaking, they will plump back up to their normal size and maybe leaf out and flower. They also make the most of whatever soil nutrients are available. Often the only nitrogen they get is when a passing coyote takes a leak within a few feet or a lizard burrows under them and dies. If you mulch it with dried bugs swept up from behind the refrigerator, that's probably
enough organic matter to be adding. The chitin in the bug's exoskeleton will break down ever so slowly, with the infrequent waterings, and that will be enough to keep the cactus happy.
If you are interested in identifying what type it is, this
Wiki page has a lot of pictures and you can maybe narrow it down.