Can I apply for a visa not at the embassy where my home university is located? Also, may I have my COE delivered to my home address?

Yes.  There is a section on our online application form where you fill in the location of the embassy where you plan to apply for a visa,  so you can fill out the embassy information in the section.  
However, if you change the place where you apply for your visa from the place you filled in the online application form, you MUST report to us immediately.  
We need to change the information on the COE application and report it to the immigration office.  Changing the information cause delay on issuing COE, so your prompt action is necessary.
We usually send COE to your university coordinator.  If you have a special request, please contact us or have your university coordinator contact us.   
If you couldn't find an answer to your question or concern, you may inquire us through the questionnaire form we shared when sending you the Letter of Acceptance or the questionnaire form shared during the orientation. If you are not applicable to these, inquire from here.