The answers are as follows:
1. This depends on your school. Predominantly, schools prefer email communication with the evaluators (process you mentioned is correct). However, most schools have a provision for a physical letter based recommendations as well. But, please confirm it with each of your selected schools.
2. Again, most schools will insist on a professional email as personal ids are considered less authentic. As for the problem with your organisation wrt mails, my recommendation to you will be to get the process started early. Give the official ids and see if they reach. If they don't, the admissions team will support you as you have a legitimate problem. Extra time in hand will help.
Hope it was useful
MBA Admissions Consultant
Entrepreneur & Educationist
Alumni, Indian School of Business (ISB), Hyderabad (GMAT: 760)
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