What Christ did for me and all of humankind

I learn about Christ's suffering from the following scriptures in the Bible that describe it well.
Matthew 26:37," ...he began to be sorrowful and very heavy." vs 38, "...my soul is exceeding sorrowful, even unto death." vs 39, "O my Father, if it be possible, let this cup pass from me." He prayed this prayer three times.
Mark 14:33, "..began to be sore amazed (astonished) and to be very heavy (in anguish)." vs. 36, "Father, all things are possible unto thee; take away this cup from me."
Luke 22:44, "and being in agony he prayed more earnestly: and his sweat was as if it were great drops of blood falling down to the ground."

Additional Insights to the suffering of Christ can be found in the Book of Mormon:
Mosiah 3:7, "He shall suffer temptations and pain of body, hunger, thirst, and fatigue even more than man can suffer except it be unto death; blood cometh from every pore, so great shall be his anguish for the wickedness and abominations of his people."
Alma 7:11-13, "He will suffer pains and afflictions and temptations of every kind...and he will take upon him death that he may loose the bands of death which bind his people. He will take upon him their infirmities that his bowels may be filled with mercy- that he may know how to succor his people in their infirmities."
D&C 19:16-19, "I, God have suffered these things for all, that they might not suffer if they would repent; suffering caused God the greatest to tremble because of pain- to bleed from every pore, suffer both body and spirit."

"To what extent was the Atonement completed in the Garden of Gethsemane"?
In Gethsemane Christ was alone. The Fathers' presence withdrew leaving the Savior to complete the victory over the forces of sin and death.

"And being in agony, He prayed more earnestly"
He prays with more intensity and becomes closer to God. He suffered the sins of the world, an unimaginable pain he bore. A burden we cannot comprehend.

"And His sweat was as it were great drops of blood falling down to the ground"
We cannot imagine his pain and agony. Physical, mental and spiritual pain that caused him to bleed from every pore. He overcame this supreme contest with the powers of evil. He took on the burdens of all mankind from Adam to the end of the world.

"Then cometh Jesus...unto a place called Gethsemane" How does the meaning of Gethsemane add insight to what the Savior experienced there?
Gethsemane means "oil press". A place where they extracted oil from olives. Similar to the blood that the Savior spilt for all of us there. It was also a place he would go to, to pray and to talk to his disciples.

What were the Saviors disciples doing while He was praying?
They were sleeping instead of watching and praying.

What had the Lord taught in Matthew 26:41?
Watch and pray always lest ye fall into temptations.

What does his example teach about how to be obedient even when "the spirit indeed is willing, but the flesh is weak"?
We can overcome our weaknesses of the flesh and do the will of the Father just as Christ did.

In D&C 10:5, how can we apply it to what we are learning?
We must pray always, like Christ did, so we can come off conqueror and conqueror satan and escape his temptations.

When we read Mosiah 15:7, 3 Nephi 11:11 and D&C 19:19 what motivated the Savior to "drink the bitter cup"?
He wanted to glorify God and allow His will to be done.

In Matthew 26:39, Mark 14:36 and Luke 22:42 the Savior said "nevertheless". What are my thoughts about that?
The Savior initially asked for the cup to pass but each scripture states how he said "nevertheless" thy will be done. He gave up what he wanted for what God thought was best for him. I need to be willing to do the same thing, trusting that God knows what is best for me.

What does it require to submit one's will to the Father in this way, no matter how painful or difficult the outcome may be?
We must follow Christs example and be ready and willing to accept and do the Lords will in our lives. There have been hard things I've been asked to go through and I always come out a better person for allowing the Lord to lead my life. He makes more of my life than I could ever dream up of having!