3) If the designated control operator does something illegal, who is responsible? The control operator and the trustee jointly.
Although both are responsible, the trustee--as the licensee--is the person who would ultimately be held responsible, and not the control operator. It should be for that reason that this was mentioned:
"It would be a good idea for the trustee to keep a record of what control operators were designated at the different times and from what location they were operating."
4) Are there any requirements of the contest or whatever you are operating that would preclude operation from multiple locations?
No, not at all, as long as the trustee allowed it. Keep in mind, however, that that is why not too many trustees will allow such things.
One thing to consider, though--the 'Thirteen Original States' event used a special event call sign, not a club call sign. The name of the ham applying for that callsign is considered the 'trustee' for it during that event. He would be held ultimately responsible for any infractions of any stations using that callsign.
On the other hand, the FCC generally is not inclined to issue fines or forfeiture orders for simple infractions of the regulations. You may well get a nice card from an 'Official Observer' stating that you've did a no-no, and asking you to be more observant of the rules in the future, however.