Some clarifications:
<< Christians believe that only baptised people have a chance of going to Heaven >>
Catholics believe that you need to be baptised to go to Heaven unless: A) Baptism was not available (people who never heard of Christianity, but were genuinely good) or B) Despite not being baptised with water, your life and heart were good enough that you are considered "baptised by desire" - i.e. if you knew, you would have been baptised... etc
Actually there are several ways, but it boils down to this:
Unbaptised people CAN still get to Heaven, but it is incredibly more difficult. In the end, though, no one can say but God.
<< and at least for Catholics, you will automatically go to Heaven if you are baptised on your death bed and say you are sorry for whatever bad you have done. >>
If you truly repent and confess, now or at the end of your life, you are forgiven through the sacrament of Confession. Baptism is a sacrament that when it is recieved also absolves you from sin, ONLY if you are truly sorry, but Baptism also removes the punishment for past sins to that point. So if given to a sincere and truly repentant person, even on the deathbed, it absolves thier sin and removes the punishment for it to that moment. Any sin after that still counts, but if the person then dies without sinning after that late Baptism they could well go straight to Heaven, yes.
Why, if they are truly sorry, would God refuse to forgive them???
<< Babies who die before being baptised (and non-Christians who are good) are believed to be in Limbo for the rest of eternity. >>
No - they are there until the Final Judgement - the end of the World. At that point, and we are not sure how, they are given a chance to freely choose God or not - the same choice the rest of us have here on earth our entire lives.