I am not sure what you mean by part of your question "begging for repentance" We are never told to beg or plead for any part of salvation as neither gets us saved. A person can plead and pour out tears over and over but until they repent and and come to faith they remain lost.
"Can a person cry out to God to save them, desiring a changed life spiritually, begging for repentance and salvation, knowing ones lost state, and believe all that Christ did for themself upon the Cross is the only way of Salvation for which they are pleading, and yet be rejected by God, and He not save said person?"
It is us who needs to repent as that part is in our hands to decide about. We are not told to beg God for repentance. However assuming that your wording is just semantics and the person has come to repentance towards God and faith in the Lord Jesus Christ, no ,God would not, and could not reject them less He reject His son and yes they would be saved and it would be seen in a life change.
As to your question "How would you help and counsel said person?"
I never try and convince anyone that they are saved when they have doubts. I take them to the word of God explain it and ask them if this is what happened in their life and let the word by the Spirit convince them one way or the other.
There is only one way given in the bible to know if we are saved and that is if we keep the commandments 1John 2:3.
I have used these messages many times and it has set some free and other it is showed them the truth as to what they need to do. I have had some leave with excitement and some tell me they are not ready to go that far. Either way the truth is known.
http://www.gerald285.com/web_media/TrueChristianityJohnMacArthhur.wma
http://www.gerald285.com/web_media/BirthmarksofaBeliever.32.mp3
One last thing I assume that everyone or certainly most has doubted their salvation at some point. However this example sounds chronic or at least an on going thing. I would not let this person teach or hold any position in the church until they had assurance as such positions only tend to add to the doubt as satan tends to attack those people with greater persistence.
Do you believe that a true Christian can have real douts of being saved?
That they can be fighting some sins on a daily basis?
that unless they overcome all known sin areas, and have no doubts, they need to question their salvation?