You can register and order from another state in your state of residence at anytime after it is issued.
You will not be filing a change of venue after litigation is completed. This is something that occurs at the initial stages of the case, not after it is has been completed.
If an issue arises later on down the road with respect to custody it is likely that Florida will be the proper forum at that time.
There is a uniform act that most states have adopted that dictates where jurisdiction is proper in cases of child custody. Normally there is what is considered as a