When a question is migrated, is the reputation earned by its author on the original site subtracted from the author's total? What about votes? Badges? Do the same rules apply to the question's answers?
Answers / Reputation
The answers on the migrated questions are deleted from the source site. This means, as with any deleted answer, the reputation will be removed.
Question and answer votes are preserved on the destination site and are cast by the Community user. These migrated votes will confer reputation.
If the user on the origin site does not have an account on the destination site, the reputation is held for that user until they do create an account on the destination site (so long as they use login credentials which are connected to their other Stack Exchange account).
Note that the above only applies for reputation earned from votes and accepts: reputation from bounties does not migrate. If a question where a bounty has been awarded gets migrated, the answerer will lose the reputation earned from the bounty, and it will be refunded to the bounty owner once the migration stub is deleted. *
The votes will earn you badges on the target site. Badges are never revoked from any user, however it does mean that it won't count toward the "next" badge of that type (e.g. if you've asked 30 positively-scored questions on separate days, and one of your positively-received questions is migrated to another site, you won't lose the Inquisitive badge, but your score toward the Socratic badge will go down to 29.)
(some of this may change a bit over time but that's how it is now)
* (If the question is migrated more than 60 days later by a Stack Exchange employee, the answerer will retain the reputation provided their answer scored at least +3 and it's been visible for at least 60 days; in that case, the bounty rep won't be refunded.)