Can change to working visa while/from abroad?

Hello everyone,
Visa and immigration related problems in Japan sure are confusing. I hope you can help me out.

I am currently under a Student visa which will end around the middle of April. I recently had an interview at a Japanese company, and although I didn't get a final answer yet, I think chances are high that I will be accepted soon.

In case i'm actually accepted, do you think it will be possible to apply the necessary documents in the immigration and return to my homeland for a vacation while they are being processed (anyway I will return to Japan before my student's visa is over), or should I stay in Japan? From your experience, how much time does it take to make this change of status? Do they issue a new CoE?

Another option, if possible, is for me to apply the documents and ask for a change of status in the Japanese embassy in my home country. Is that even possible?

Thank you for your time!

Tom

In case i'm actually accepted, do you think it will be possible to apply the necessary documents in the immigration and return to my homeland for a vacation while they are being processed (anyway I will return to Japan before my student's visa is over)

It is possible, but you must return before your status expires. If you want to return during the two-month "grace period", you must discuss it with immigration beorehand, as it is possible they will tell you to apply for a re-entry permit, or even that they will not allow it.

Do they issue a new CoE?

No.

Another option, if possible, is for me to apply the documents and ask for a change of status in the Japanese embassy in my home country. Is that even possible?

No.

Technically speaking
COE application is to facilitate your visa application at Japan Embassy in your country.
COE is applied on your behalf by a sponsor/someone in Japan.
COE application in Japan, visa application at Japan Embassy.

Change of Status resident is only done inside Japan at immigration office. No COE involved.
Your current status stay will be extended once application is accepted. Extension up to 2 months or application decided.

As for whether you can travel abroad while your Change of Status is pending, since yours is of student status , should be OK.
Double check with immigration to confirm.

Thank you so much for your reply! Helped a lot.

Tom