When the nuclear commander of the person who is playing Havoc dies, can he still play the free card?     

Yes, he can (This is an official WOTC FAQ answer).

Can Mind Control negate the Havocs use of a free nuclear card?

It can prevent the use of any card, even one used has part of an ability (This is an official WOTC FAQ answer).

Does Wartime Inflation alter the cost of a card for Mind Control?     

Wartime Inflation would alter the cost of the Mind Control, you have to pay the cost of their card, plus the higher cost because you are playing a card (This is an official WOTC FAQ answer).

If Wartime Inflation had been played will the humanitarians faction have to pay 1 energy for their free card or are they unaffected?     

The humanitarians get their card still for free. They are not required to pay '0 energy' for it in the wording, so they still get it just the same for free since their is no actual cost associated with it (This is an official WOTC FAQ answer).