The Father does not "decide" any such thing. Paul is very careful how he words his teaching in Romans regarding this and makes it clear that the vessel of wrath becomes such by the decision of the person himself. God would not need to "patiently put up" with someone He had determined to make that way!That He's decided that you are better suited to be a vessel of wrath, rather than mercy?
Can a person be saved whenever they want? A misleading question. A person can be saved whenever the Spirit opens their heart to accept the message of Christ, as in the case of Lydia in Acts 16. She obviously could not have been saved without knowing about Jesus.
Is there a possibility of salvation after death? Scripture is crystal clear: no.