The correct answers are I must have stepped on a butterfly and altered the course of history and I will travel to 1920, and my assistant, to 1930.
I must have stepped... and altered...
This is a correct example of subject and auxiliary ellipsis. The subject "I" and the auxiliaries "must have" are successfully shared between the two coordinated verbs ("stepped" and "altered").
I will travel to 1920, and my assistant, to 1930.
This is a flawless example of gapping. The verb phrase "will travel" is perfectly understood and omitted in the second clause.
Incorrect options:
"...and my assistant has, the temporal maps" is ungrammatical. You cannot leave the auxiliary "has" stranded right before a new direct object when gapping the main verb.
"...and my assistant did too" is ungrammatical. The first clause is in the passive voice ("was chased"), but "did" attempts to substitute for an active verb phrase. They do not match.