What does Jesus mean when He says, "for unless you believe that I am He?"
The controversy surrounding the Lord Jesus Christ is whether he was, in fact, the "Son of man" (notice that Jesus uses "Son of man" at John 8:28 and "Son" at John 8:36, both referring to his title at Daniel 7:13) that is to say, the promised Messiah, or "Christ," that the Jews had been expecting, "the prophet that was to come" (John 6:14, 7:26). You should also notice that in this same verse -- John 8:24 (from which you have quoted) -- and at John 8:28, the Greek words _egó eimí_ are rendered in both verses as "I am he," while at John 8:58, the same Greek words are rendered as just "I am," without the addition of the word "he," evidently for theological reasons to make it appear that Jesus uses the title, "I AM" at Exodus 3:14.
Many of the folks that make this theological argument do not know that the Greek Septuagint doesn't use the words "_egó eimí_" at all, but instead renders this specific portion of Exodus 3:14 as _ho On_, which means "The Being" or "The One who is," and _ho On_ ("the One") is applied to Jehovah at Revelation 1:8 and at Revelation 11:17.
Of course, there are some folks that do know this, but they have an agenda. When you get to John 8:56, Jesus is still making the point that he is the Messiah, for he says there that Abraham's faith led him to rejoice in the prospect of the arrival of the promised Seed, the Messiah, for at Genesis 22:18, we read God's promise to Abraham that it would be "by means of [Abraham's] seed" that all nations of the earth would bless themselves, Jesus also indicated at John 8:58 that he had been in a position to have observed Abraham's faith in action since in Jesus' prehuman existence, before Abraham had not even been born, Jesus had existed.
Similarly, Jesus takes on the attributes of his Father and sometimes the functions too.. but he is a "creation" both in a heavenly and earthly sense. So, No, he is NOT Jehovah.
Adding to what you have said here, Jesus is referred to at Revelation 3:14 as being "the beginning of the creation by God," so not only is Jesus God's firstborn, but he is God's first creation, who the apostle Paul states, at Hebrews 10:12, 13, had been biding his time sitting at God's right hand "until [when] his enemies should be placed as a stool for his feet" (Psalm 110:1).
However, it is when the end of the "appointed times of the nations" (Luke 21:24) had arrived that God "again brings his Firstborn into the inhabited earth" (Hebrews 1:6) to rule "in the midst of your enemies" (Psalm 110:2) during his invisible presence after having girded on power in his having dealt a blow to his heavenly enemies in 1914 when Jesus began ruling as king (Revelation 12:7-12), and having received an anointing from God as a prophet, as high priest, and in his kingship.
Paul distinguishes Jesus from the rest of God's angels, for Jesus is the only one of the angels to whom God refers as "my son," underscoring the fact that Jesus' anointing was "more than [had any of Jesus'] partners" received from God, their anointing having only been as a king (Hebrews 1:5-9).