Valid work visa but changes in work situation

Hello everyone,
I was working in Japan for a big Japanese company for one year and a half and officially quitted end of August. The reason was mostly that I couldn't handle the rythm anymore (service industry) but also the work ethic and heavy atmosphere in general. I was a very valued member and so they kept pushing me and saying that I could do some other sitting jobs, or jobs abroad, even back home. Pressure was big so I just ended up telling them I need to quit because of my health (I got injured as a teenager and lately got worse because of all this standing work), so I wanted to focus on getting better. I didn't tell them I also wanted to quit because I didn't want to work for them anymore and therefore, didn't say anything about searching another job. They think I just quitted to get better. It's not something I'm comfortable with but otherwise, precedures to quit would have been endless and the most important was to protect myself.
Anyway, that was just for the background. I still have 2 days to notify the changes to the immigration office and before that, I had some questions to ask and couldn't find the answers anywhere.

First is, in Japan, do former companies have to notify the immigration that their sponsored worker no longer work for them? I still have 3 years and a half of valid working visa (originally 5 years) in Specialist in Humanities/International services.
My first plan was to search for an other job directly. In doing so, will my old company be notify by the immigration that I'm searching for a job (as they sponsored my visa)?

I also have a second plan, which suits me best (I'm still young and not sure if I want to jump straight in a new job) but I'm not sure I can do with my current visa. I would like to apply for an intership, in order to gain more knowledge and experiences in a different field. I would combine this with a part-time job to get a small income in order to provide for my day-to-day life here (with some help from my family the time of the internship). Would my visa authorized it? The nternship shouldn't last more than 2 or 3 months. And secondly, is there a minimum working hours for a work visa?
Or do some of you think that I should just not say a thing to the immigration, do as I like with my visa, and then get a job hopefully in 3 months and just notify the immigration then?

I know it is a lot of questions but I would really appreciate some helps, as the clock is ticking since I only have 2 days left to report employment changes to Immigration.
Thanks in advance.

First is, in Japan, do former companies have to notify the immigration that their sponsored worker no longer work for them?

Yes. They have to report the termination of the contract just like you do.

will my old company be notify by the immigration that I'm searching for a job (as they sponsored my visa)?

No. Besides, how would immigration know that you are searching for a job?

Would my visa authorized it?

You are forbidden to engage in any paid activity which does not fall within the scope of your status without prior permission (which in your situation will almost certainly not be granted).

Thank you very much for your answer Firas.

As I was affraid, I cannot engage in any intership activities without a change in my visa status then.
But I also don't want to "waste" my work visa, so I guess I better notify the immigration and find a job in the field that interest me within 90 days.

Is there a minimum work hours with a work visa though? I guess part-time is out of the question but are 30-35 hours/week considered not enough?

Thank you again for the quick reply!

There is no formal minimum work hours requirement, but typically you must earn enough to make a living, which is difficult if you don't work full time.

You do need to inform immigration that you have left your job,though, within two weeks, just like your former employer has to. You can tell them you are looking for another job, and when you find one, you can report the new employer to immigration again.

Thank you so much for your answers you too.

I'll get going to the Immigration tomorrow first thing in the morning.

NO need to go in person, you can do it by mail or online. http://www.immi-moj.go.jp/english/tetuduki/kanri/shyorui/19.html

Hi hannah, i was in the same situation as you. Please send me a private message and i am going to tell you more details.