Re: F2 visa

Ive recently received my F2 visa and I want to confirm exactly what I can and cant do. My understanding is that I can legally work at any hagwon/school without needing to register with immigration (part-time and/or full-time)...is this correct?

Also, regarding privates - am I allowed to legally do privates now, and do I need to register this somewhere? I assume I would need to be paying tax on this income.

Finally, if I was to work full-time at a hagwon, is there any need to be signing a contract? I have been reading through some posts on this forum and am under the impression as an F2 visa holder, you are not offered the usual deal as an E2 (rent, bonus after 1 year, flights etc) and you would negotiate your salary by the hour/class. Is this the norm?

Thanks in advance for any assistance you can give.