Thanks for the question and sorry we didn't get back to you sooner. I've been finishing an update to a Web app I have:
Holy Quotes from AskACatholic
I now have published several web apps you can download for free on Androids. Since I don't have a smart phone, feedback is greatly appreciated, that addresses the question: How do they look?
For a full and complete understanding of my answers to your questions, I highly recommend you buy a cheap copy of the Catechism of the Catholic Church. I can answer a good portion of your questions but only the Catechism will give you the full answer you deserve.
I am having a hard time understanding what Jesus actually did in the world.
- [Is|Was] Jesus' crucifixion was part of God's plan?
What I mean is:
- Did Jesus need to die on the Cross or could He have redeemed mankind in another way?
Yes, it was part of God's Plan; a Plan His Son, Jesus, willingly accepted.
On the issue of whether Jesus had to die on the Cross to redeem our sins, Yes, but He did this in accordance with the Father's Plan to redeem mankind. Now one could also ask:
Could the Father have chosen a different way? <Sure! He could have.>
But He did not.
- Finally, if Jesus' crucifixion was God's plan, then was Judas' betrayal and the betrayal of those people who asked Jesus to be crucified, also a part of God's plan?
No, Their choice [Judas and those who betrayed Him] to betray Jesus was not part of God's Plan. They had free will to choose to be with the Lord's Plan of redemption, including the Jewish people many who, today, have embraced Catholicism and call themselves Hebrew Catholics, or not to be with His Plan.
Those Jewish brethren who have embrace the Church, believe they haven't renounced their Jewish-ness but have fulfilled it as a Catholic.
- So what is the difference between Jesus' Will and the will of those who betrayed Him?
- Jesus' Will was always one with the Father though they (the Father and the Son) are two separate Divine Persons of the Trinity.
- The will of Judas and those who betrayed the Lord was a weaker will due to Original sin.
Again, pick up a copy of the Catechism for a fuller, more complete answer.
My colleagues will chime in if I have missed anything of importance.