Before Abraham WasI AM

When did they pick up stones?
When Jesus said"Before Abraham wasI AM" (John 8:58).

Why did they pick up stones?
To stone Jesus to death.

When would you expect them to be angry enough to try to kill Jesus?
When He called them the sons of the devil (see Your father is the devil).

To themwhy was Jesus declaring "Before Abraham wasI AM" worse than calling them the sons of the devil?
God stated “I AM” as His name in Exodus 3:14: Then Moses said to God“Beholdwhen I come to the children of Israel and say to them‘The God of your fathers has sent me to you,’ and they say to me‘What is His name?’ what shall I say to them?” And God said to Moses“I AM WHO I AM.” And He said“Thus you shall say to the children of Israel‘I AM has sent me to you’” (Exodus 3:13-14). Jesus was declaring in terms crystal clear to them that He is Godand this declarationwhich they mistook as blasphemyis what enraged them to try to kill Him.

Then why didn't the Jews consider it blasphemous when Jesus said "I am" to them earlier in this discourse as recorded in John 8:24 and 28?
As mentioned in Die in your Sinsthe original Bible recorded in Greek Jesus' discourses with Jewish audiences in Aramaic. The originally recorded phrase for "I am" in John 8:24 and 28 is εγω ειμι (ego eimi)which is the Greek equivalent of the originally spoken Aramaic phrase, אְנָא הוּא (ena na). The originally recorded Greek for "I AM" in John 8:58 is also εγω ειμι (ego eimi) but the Jews suddenly flying into a murderous rage upon hearing it indicates that something was different about this spoken phrase than those recorded in John 8:24 and 28. The only logical explanation is that Jesus had switched to Hebrew and said אֶֽהְיֶ֖ה (eh yeh)which is equivalent to εγω ειμι (ego eimi) in Greek, "I AM" Englishand is also the exact Hebrew name that God used to identify Himself as "I AM" in Exodus 3:14 above.

How did Jesus hide Himself and go through "the midst of them" (John 8:59)?
The same way He did when the people tried to kill Him upon hearing His very first sermon in Nazareth: "And hearing these thingsall in the synagogue were filled with rage. And having risen upthey thrust Him out of the town and led Him to the brow of the hill on which their town had been builtthat they might throw Him down headlong. But passing through the midst of themHe went His way" (Luke 4:28-30).

How can God make Himself invisible and walk through a crowd?
 

What did Jesus mean by “Your father Abraham rejoiced that he would see My dayand he saw it and was glad" (John 8:56)?
Abraham saw Jesus and was glad.

How could Abrahamwho lived two thousands years before Jesus was bornhave seen Jesus?
Jesus appeared to him with God the Father and even spoke to him. Read the explanation below Genesis 15. Note that Abraham is still called "Abram" in these early chapters of Genesis.