Working as direct-hire ALT, want to do side gig – kojin jigyo? Taxation? Help! (Xpost /japanfinance)

Hey everyone, hopefully I can get some guidance here. I searched for similar questions and found some situations *close* to mine but not exactly analogous.

I currently work full-time for a local city government as a direct-hire ALT, Monday to Friday, 35 hours a week. Already enrolled in health insurance (me and my daughter, wife is covered under her own as she works full time) and national pension. I also do a conversation class on Friday nights at a per-lesson rate. I am on a spousal visa and so my work is unrestricted, no problem there.

I’m going to be taking on another side job, working for a startup based in Japan. Initially doing a sort of piece-work/freelancing arrangement, but maybe transitioning to full time in the future.

The income for both of these jobs is not likely to exceed my current main salary. I wanted to know if I should be doing anything extra in terms of paperwork to make sure everything is on the level. So, a few questions:

* Do I need to make a kojin jigyou, or file for one, or however that works? Or should I be considered a gyomu itaku worker for the time being?

* When it comes time for filing taxes, would this income be filed as ‘miscellaneous income’?

* How do I make sure that my earnings are properly reported for the purposes of residence tax?

* As I’m already enrolled in health insurance and pension at my full time job, I don’t have to do anything additional, correct?

* Basically, what are the steps I can take as an individual to make sure there are no issues or problems with this work vis-a-vis taxation and legality?

Please feel free to ask for any clarification you need. The startup that I’m going to work for is a very small company and I would be their first employee. They’re still navigating starting the starting of the business themselves so they might not have the best answers for how to arrange things.