Can I divorce my wife without going back to her home country? Can I re-marry without getting divorce?

You don't say where your wife is, if she is an American citizen (or if you are, you only say you're a "national"), if you have kids together, or really provide enough information to answer your question. However, you may generally obtain a divorce in FL if you meet the six (6) month residency requirement, and provide proper notice of the divorce action to your wife by service of process. You cannot re-marry someone else in the United States without first obtaining a divorce in your current marriage.

Yes you can file the case in Florida. No you cannot get married until your current marriage is ended with an official divorce decree.

You cannot remarry without first getting a divorce. You can probably file in Florida and try to have her served in Africa. You might have to publish in the paper if service cannot be accomplished. Hire an attorney in your area. Florida has jurisdiction over your divorce if you have been a resident for at least 6 months.