According to what the teachers here said, 1) is right and 2) is wrong.
1) The bus leaves at 5:00, reaching the destination at 6:00. (o)
2) The bus leaves at 5:00, then reaching the destination at 6:00. (x)
3. The bus leaves at 5:00, and then reaches the destination at 6:00.
That one is correct. For some members it would be acceptable without the 'and'.
Your sentence (2) is not correct, for the same reason (4) is not.:
[4)] The bus leaves at 5:00, and reaching the destination at 6:00.
I like 5jj's explanation of why (2) is incorrect. The explanation is based on the assumption that "then" can be a coordinating conjunction and is one in (2).
The incorrectness of (2) may also be accounted for on the adverb interpretation of "then," which yields a meaning that is impossible in the world as we know it.
On the adverb reading of "then," (2) would mean that the bus leaves at 5:00 and reaches the destination
at that time (5:00), which is said to be simultaneously 6:00!
Consider the following sentences:
(a1) The bus left at 5:00, then having a flat tire. [It didn't get very far.]
(a2) The bus left at 5:00, at that time having a flat tire. [It didn't get very far.]
(b1) The bus left at 5:00, having then a flat tire. [It didn't get very far.]
(b2) The bus left at 5:00, having at that time a flat tire. [It didn't get very far.]
As you can see, "then" (on its adverbial interpretation) does NOT yield a "subsequently" reading in modifying the participial phrase. "Then" specifies the time of the situation expressed by the participial phrase, identifying that time deictically with the time of the bus's leaving (the situation of the main clause).