Sickness leave / Allowance / Rules

Hello everybody, I am seeking help to understand some things relative to job, sickness leave, shakai hoken, rules..

First of all I am going to quickly describe the context : I joined this company on June 2025 and became a full time employee on August the job is far from where I live (around 2h far) but I did not have a choice at that moment I needed to make a living. As time passed, my mental health took a great blow, waking up at 5am and coming back at 9pm sometimes more. I took the hit but it was becoming harder and harder for me and my wife, I was a walking zombie, stress, anxiety, and sleep troubles. I decided to get help from a Psychiatrist, he advised me to take 1 month off with a Medical Certificate, at first I did not take it but I took it after all because my health kept deteriorating.

I did not discuss with my company as I consider my health a private matter, the Psychiatrist ordered me to take a one month off work. Of course the boss was not very happy about the sudden news, and he kept pressuring me to reconsider and also gently threatening me. He told me that since I did not work 1 year yet I could not take days off (but it is literally not normal days off) and I would not receive any financial compensation, but if I am not mistaking, sickness allowance is a right I have as a full time employee whether I have worked 6 months or 1 year. He also can not refused to fill in the sickness allowance paper right? Can some of you confirmed this?
Also he is threatening to fire me considering I have broken some company's rules (really I dont now which rules he is talking about). But I did not do anything against the law, it's not like I intentionally gave up and took one month by myself, it was by doctor's order and I have a medical certificate. Does he have the right to fire me over a sickness leave ?

I am already very stressful and anxious, with a boss talking to me like he owns me it gets more and more difficult.

Another question : If I ever decided to quit on Doctor's advice (so woth a medical certificate), will I get financial support from Hello Work during the time I'll look for another job?

My japanese wife and family never had been in this situation so they don't know about anything related to my problem.

Can someone relieve me and lift the some weight of my shoulder?

Thank you for your future answers and helps.

by Low_Buy2248