Yes, you can.

Students who have graduated from a caregiver training institution between the academic years of 2017 and 2021 (before the end of March 2022) who have not passed the National Caregiver Certification Examination (due to insufficient marks or absence) can apply to the Social Welfare Promotion and National Examination Center to register as a caregiver for a period of 5 years and receive the certificate necessary to apply for  the "nursing" visa.

They must, however, a) pass the National Caregiver Certification Examination within 5 years after graduation; or b) extend their caregiving certificate if they work continuously for 5 years within the care services field.

For more information, please access the following PDF: http://www.sssc.or.jp/touroku/pdf/info_keika_2018.pdf

You are allowed to have a part-time job.
First receive permission at the Immigration Bureau to engage in activity other than that permitted under your current status. If permission is granted, you can work 28 hours per week part-time. 
During the long holidays set by the school, such as the summer and spring holidays, you can work up to 8 hours a day (40 hours per week).

It is not necessary.
However, it is desirable to have around N2 or higher when entering a school.

Yes, registration is possible.
Students who have graduated from a caregiver training institution between the academic years of 2017 and 2021 (before the end of March 2022) who have not passed the National Caregiver Certification Examination (due to insufficient marks or absence) can apply to the Social Welfare Promotion and National Examination Center to register as a caregiver for a period of 5 years and receive the certificate necessary to apply for  the "nursing" visa.

They must, however, a) pass the National Caregiver Certification Examination within 5 years after graduation; or b) extend their caregiving certificate if they work continuously for 5 years within the care services field.
For more information, please access the following PDF: http://www.sssc.or.jp/touroku/pdf/info_keika_2018.pdf

You must apply to the Social Welfare Promotion and National Examination Center. After examination and registration, your registration certificate will be issued.
It normally takes one month from the time of submission to issuance. (If there are deficiencies in the application form or papers, it will take one extra month after resolving the issue)

In accordance with the law, you must pay a registration license tax and registration fee.

You can start the application procedures if you have been granted permission with your current status of "Designate Activities (care worker candidate)". You can apply for the status change under the condition that you will later provide the caregiver registration certificate. 

To be able to change your status to “Nursing”, you will need to have a registration card. You will not be able to change it until you acquire your registration card. However, until you change to “Nursing”, if you are able to change your status to “Designated Activities”, you will be able to engage in full time nursing work after April 1st.

Yes.
You need to complete the required paperwork at the Immigration Bureau to get the "Designated Activities (care worker candidate)" status until your official status changes. After obtaining permission to engage in an activity other than that accepted under the status of residence, part-time work will be allowed.

You cannot change your status to "Nursing" even if you have graduated from a Japanese Language School.
In order to get the "Nursing" status, you need a Graduation Certificate from a caregiving training facility and a Caregiver Registration Certificate.