Can I request court to order alimony and, if alimony (let us say $500 per month to me) is ordered by the court, still can we both (me and my husband) continue as wife and husband without ever getting divorced?
You can request anything you like. And, as I explained in my prior response, section 61.09 of the Florida Statutes does, in fact, allow for a spouse to "apply to the court for alimony . . . without seeking dissolution of marriage."