It looks like @lazar has already answered most of your questions (thanks Lazar!) but I wanted to quickly address your first question about whether to apply to join Crossref under the name of your publication or your publisher.
It is always preferable to apply under the name of your publishing organization, in this case your university. As a Crossref member, you will deposit metadata/register DOIs for all of the materials you publish, so initiating a membership under the name of your parent university allows you to manage all of the university’s publications through a single account (and only pay a single annual membership fee, if you choose to apply for independent membership).
The only reason to apply under the name of a publication would be if you had no parent publishing organization or if your journal were published by an individual. We cannot set up Crossref accounts in the name of an individual person, so in these cases only you should apply under the name of your journal or other publication. However, this does not sound like it applies here @cugmreditor1 , so you should just go ahead and apply in the name of your university.
If you have any other questions please don’t hesitate to ask.
You need to contact the official DOI via email, and there will be a form link for you to fill out. Registering now only requires a membership fee of six months, which seems to be just $125! The cost-effectiveness is very high, my brother. What academic search is your university journal indexed by?
Please specify the geographic region to which the publisher belongs so that we can discuss the possibility of helping you obtain Crossref membership We are a sponsoring organization in the Middle East and North Africa