Can't speak to blood loss, but can speak to the suicide attempt stuff. I was hospitalized repeatedly for suicide attempts as a teen (ugh) and some of those experiences apply here.
So, the questions you're asking depend HUGELY on the individual hospital, the character's state of mind when he wakes, and state law. Family will be with him in the ICU; mental health professionals will ask them to leave during assessments (interminable and repeated Q&A sessions where the psychologist/therapist/social worker asks a long list of questions about depression/substance use/self harm/thoughts of harming others/etc) but otherwise they're welcome, in my experience. Same as with any injury. In a psych ward, family or friends can visit during visiting hours, and he can call them on the phone if he has a phone card.
"Suicide watch" is a pretty vague term. In the ICU, there will be nurses around more or less all the time, and the patient will be in bed, so there doesn't need to be any specific "watch" happening (I expect the nurses are aware, but I was never treated specifically differently at the ICU stage). If you're talking about within a psych ward, everyone in a psych ward is assumed to be potentially suicidal all the time. Nobody gets to keep their belt.
Basically, while the patient is in the ICU, unless they're actively psychotic or otherwise dangerous (sounds like your protag is just depressed) they are treated like any other injured patient. Once they're in the psych ward, they're treated like any other crazy person.
Exception: I was in one hospital where after the ICU but before any question of discharge/psych ward, while I was recovering in a single room, they had a staff member assigned to sit with me around the clock and just keep an eye on my mental health. That was an excellent hospital, however. When I was completely physically recovered (after several days), they did a psych eval on me, and since I REALLY did not want to go into the psych ward, I smiled prettily and asked to go home and they were fine with that. Psych wards have very little space and aren't there to make you happy and well again, they're just a box to stick you in to keep you alive for a few days while they get your meds sorted. If you aren't in immediate danger of killing yourself or someone else, they'll send you home.
By comparison, the worst hospital I went to had me in ICU for a few hours and then sent me to the psych ward while still unconscious. I woke up alone and too dizzy to walk, in a single room with a shower. Huge variations in how much care a hospital puts in to its patients.
There are some more layers with voluntary/involuntary commitment. Short version: Big hospitals usually have a psychiatric unit of their own, which is for short-term care. They'll have a Pediatric Psychiatric Unit and an Adult and Geriatric Psychiatric Unit, or some similar name; your protag will be in with the kids and teens if he does get sent to the psych ward. If you're the kind of sick that never gets better, they send you off to the state hospital. Otherwise, they sort out your meds and boot you out as soon as you answer "nah" to "anything fun planned for the weekend like for example suicide?" If you try to kill yourself but are like, "that was a poor life choice, not doing that again," they'll probably just send you home from the medical part of the hospital if you seem pretty stable. If you seem like a risk to yourself or others, they'll put you in the psych ward to sort out your meds; if you don't want to stay voluntarily, they can hold you for 72 hours (depends on the state I think) but then have to get a court order to hold you longer.
Let me know if there's any more info that might be helpful.