It really depends on the therapist. Generally they are supposed to talk to you and access if they think you are an immediate threat. Suicidal thoughts alone aren't really enough, but if the therapist thinks you're in danger, they have an ethical duty to warn.
If you say something like, "I don't think my medication is working, I'm having passive thoughts that life isn't worth living", almost 100% the therapist will ask a bunch of questions.
Generally, in my experience, if you say, "I have no immediate plan to hurt myself, I don't think the inpatient environment would help me, I want to seek other options right now" they're open to creating a "safety plan", i.e. maybe they'll say, go back and have your medication adjusted, or come to therapy more often, or talk to your pastor or something.... make some sort of change towards safety.
Self harm.... really depends. I don't want to say yes or no to either of your questions really because it's all up to what your therapist is like and what you are like. Some people cut daily and they aren't hospitalized because their doctors agree the hospital environment just makes them spiral, and their cutting is never very bad... I met several cutters back in high school like this, they would hardly draw a drop of blood and were still doing fine in classes, no reason to take them out of school, destabilize them further.... Other people cut and the second it is discovered they are sent to the ER because it's deemed very dangerous. Lots of people self-harm by doing drugs or drinking or even things like hitting themselves but their therapists don't try to get them to go to hospital.... other people, the therapist would.
I think if you open with, "I don't think going to the hospital would improve my situation, and I don't think I am in any immediate danger, but I want to create a safety plan and address recent passive thoughts of suicide", the therapist will probably not immediately want to send you to the hospital.
Even if you do get sent to like the ER for an evaluation, it's kinda the same deal. If you tell them you would rather work with your current therapist, say you'll go and see your psychiatrist soon to get meds adjusted, don't feel in any immediate danger, have no immediate plan, have someone to call if you feel in danger.... I mean, I wouldn't hospitalize that person, but same thing.... no guarantees.
