Either I don't understand what the the questioner is asking or the questioner doesn't understand what a Governor's Warrant is. A GOVERNOR'S WARRANT is a request for extradition. Individuals don't apply for them, they are requested by the State Attorney General in order to extradite a fugitive from justice from another state, back to face prosecution.
A Governor's Warrant (also known as Extradition) is issued when the apprehending state officially notifies the "wanting" state of the fugitive's arrest. It is both a legal and an administrative process, and how long it takes is not governed by statute. It can take as long as it takes. If the fugitive is still awaiting extradition in jail after 60 - 90 days, they should file a Writ of Habeus Corpus.
A "Governor's Warrant" is the formal legal name for an Order of Extradition.
There is no jail time associated with the issuance of a governor's warrant, except the time that you may spend incarcerated in the apprheending jurisdiction as you await the arrival to take you back, of the officers from the state that issued the extradition order.
If you have fled the "wanting" jurisdiction, you may be charged with being a "Fugitive" in addition to whatever the original criminal charge against you was.
When a judge signs the warrant (I assume an arrest warrant) it goes into effect immediately. However, depending on the court system, the warrant is sent to the Sheriff's Office or county police department, where it is entered into their system for statewide or nationwide viewing, depending on the severity of the crime or if it's for failure to appear. This can take some time (but not usually more than two weeks).
Depends on what the warrant is for
90 days
A Governor's Warrant is also known as a Writ of Extradition. Once the state that wants you has indicated that they will extradite you, the holding state will keep you in jail until the formal legal process of extradition has been accomplished. This is not exactly a speedy process and if you are still in jail, awaiting extradtion, after 90 days I would suggest filing a Writ of Habeus Corpus.
24 hours
It depends on what the offense is that you are wanted for.
I presume you mean a bench warrant, not a beach warrant. You can remain in jail either until a new bail is set and is posted, or until the case is resolved.
"Failure to appear" in Alabama will cause a bench warrant to be issued for your arrest. Usually, you would not be picked up on a bench warrant unless they run your ID for another reason and the warrant shows up. You will be taken to jail under the warrant and will remain there until (a) you post bond, or (b) you go to court. You will be assigned the "next available" court date, sometimes as long as 30 to 60 days out.
4 days
24 hours
You will set in jail until one your court date or two you bail out of jail. bail will depend on what you are being held for.
If no one raises your bail money you will remain in jail until your trial.
A Governors warrant is a special type of warrant and in most States there would be no "expiration". Most warrants per se have no expiration date. Once issued they are good forever (or until the defendant dies).