If you need stitches/glue you need to go iin within 12 hours usually.
If it's within my skillset to handle it myself then I do. I'm usually pretty careful about my first aid. Steri strips if necessary, Neosporin, nonstick gauze pad, medical tape.
TW for more specific descriptions of severe SH
Possible trigger:
I use steri strips myself if it's pretty far into the dermis. I try to strategically plan the locations of deeper cuts so I'll be able to steri strip them without pulling other ones open.
I get stitches if I cut into the fat, especially if it's more than just barely nicking that layer. I used to do them myself as long as it wasn't deep enough to require thicker thread or internal sutures, but I don't recommend that, especially if you don't have training/experience, and I got rid of the tools a while back so now I'd have to go to the ER for any sutures.
It'll depend on the state, the hospital, and the doc, but I only got put on the psych ward once for non suicidal self injury, and it was the first time I went in and they were pretty freaked out by the severity. After that I usually got a psych eval and was then let go because they could see that putting me on the psych ward hasn't helped. There were a few times they insisted on keeping me in the ER for a while longer when things were really bad, but didn't admit me to the psych ward.
The most important thing is keeping them clean. Disinfect your skin beforehand, put a bandage on immediately after. As long as you're not cutting near joints and you're staying away from hands, feet, and areas with tendons/ligaments close to the surface, stitches or glue are usually used so that it will heal more quickly and so that you'll have a smaller scar. I'd never discourage anyone from seeking medical attention, and if it does go into the fat then it does become more important to get medical attention.
Lastly I just wanted to pass along something they said to me one of the times I was in the ER. They always asked me to come in before I hurt myself rather than after and I brushed them off because that seemed like I'd be wasting their time. One time someone, I think it might have been the psychiatrist but I don't remember, said that they'd rather I came in before instead of after, that way they could feel like they were actually able to do something and help me rather than just repair the damage after.