Why did the stick turn specifically into a snake?
Da'as Zekenim, Hadar Zekenim (verse 2): Just like a snake bites and kills, so Pharaoh and his servants bite and kill Yisrael. 3
Da'as Zekenim, Hadar Zekenim (verses 2,7): It returned to a staff, to show that Pharaoh and his servants will be like dry wood. The Tzara'as hinted that Pharaoh and his servants will be Tamei and be Metamei Yisrael. His hand returned to normal, to hint that Yisrael will be purified from Tum'as Egypt.
Why did Moshe run away from a wooden snake?
Seforno: The snake that the stick turned into was a real live one. 1
Seeing as Moshe did not doubt that it was Hashem who was speaking to him, why did Hashem perform these wonders in front of him?
Ramban #2: Hashem did this in order to substantiate His statement to Moshe that He would appear to Yisrael with the Name of Hashem and perform with them supernatural miracles, 3 of which these were a demonstration. 4