For example: I'm playing as Mordekaiser and have a ghost of Graves, if I (Mordekaiser) get Exhausted, does that reduce my ghost's damage? Is it the same with Annie and Tibbers, or Yorick and his ghouls?
No. Minions have nothing to do with the champions stats or effects unless they are minions created by the champions abilities (excluding Malzahars pet voidling) such as Yorick ghouls which scale off his attack damage and max hp, Shaco clone which is given statistics from the original Shaco and Mordekaisers Pet of the grave which is given stats from Mordekaiser. Those are the few champions which are the exception to the rule.
Hope this was helpful.
I don't think so, that minion is made at that time so if anything happens it shouldn't affect the minion since it was made at that time and scales from that time.
In the case of Mordekaiser's ultimate and Shaco's clone, they scale off of the champions stats, so it depends on when the exhaust is cast. If it is cast after the clone is created, I'm fairly certain the clone maintains the stats it inherits when it spawned.