Why did he approach the gate of the furnace?
Rashi: He wanted to talk with them. Initially, he saw them from afar 1 .
One Perush in Rashi (Sanhedrin 92b) says that the plaster (or foundation) of the furnace melted, so the heat burned those who threw them in. We must say that the heat ceased. If not, Nebuchadnetzar could not come close! (PF)
Why did he say "Avdohi Di Elaka Ila'a"?
Malbim: This is the act of the supreme G-d. Lower natures are submissive to Him. They used to attribute them to lower gods.
Did they climb out of the furnace?
Rashi: No. He told them "leave", and not 'ascend'. We infer 1 that the floor of the furnace rose even with the ground, lest they need to toil to ascend.
Sanhedrin 92b says that we have a tradition for this, but it is not learned from the verses! (PF)
Why didn't they leave before he told them to leave?
Rashi: Even though they trusted in the miracle that will be done for them, they did not disgrace the kingship, to leave without permission.
