Can I register a business?

One doesn't register a sole proprietorship. One obtains a dba (fictitious business name statement) and they don't check immigration status.

It depends on what kind of business entity you were. If you were a sole proprietor, you wouldn't really have any status to fall out of other than letting your fictitious business name statement expire. If you were a corporation or other entity there are steps you can take if you were suspended or otherwise made inactive by the Secretary of State or Franchise Tax Board. If you want more information you will need to provide more facts in your question.

This is a confusing question, what status are you talking about. Contact an attorney for an initial consultation, they are often not very expensive.