Would I be sent to a mental hospital for these issues:?

I am depressed, I cut my right forearm, I think of suicide 20/7, I want to die 15/7, I have a plan of how I want to go, I'm bordering anorexia, I'm bullied, I'm bisexual and in the closet (not that that really matters), I'm a pathological compulsive liar, my dad is severely bipolar, 3 of my other close relatives are schizophrenic. I fear I'm one of those as well. I'm 14. I know I need help, but I don't know if my issues are enough to go to a psych ward or whatever. Please don't say I'm screwed up and should be enjoying life, because I tried...