Pitying fools as hard as I can...
The Demagogue's ability says "Instead of attacking chose a mutation from ..."
Because of the "instead of attacking" this ability can only be used during the attack phase correct?
Furthermore, can the summoner's ability allow me to have more than five cards in my hand?
Yep. There is no hand limit (you just won't get to draw if you have 5+ cards at the beginning of the turn). Fishing out a mutation to build as magic (with the intention of digging it back out of the discard pile later) is a common and effective Filth tactic.
If I take control of an enemy common unit with heretic's rebuke
that has a wound on it, will the mutation I summon carry over the wound?
What about if the mutation in case is corpulent mutant, do I need to roll I die to see if the wound carries over or it's automatically placed on the summoned mutation.
You know I'm not sure. I would assume the wounds carry over, but it's not FAQed anywhere.
it is possible to use Heretic's rebuke and channel corruption at the same time on the same unit (Play Heretic's rebuke, take control of enemy unit but instead of paying the cost of a mutation play channel corruption and place a mutation on controlled enemy unit for free). Is this correct?
Nope This one is in the FAQ: http://www.plaidhatgames.com/images/games/summoner-wars/swfa...
In case you don't feel like digging:
22.214.171.124: Can I use Channel Corruption to play a Mutation with Heretic's Rebuke?
No. You cannot play another event card until one event has completely resolved. In this instance, Heretic's Rebuke will not have fully resolved because you will not have played a Mutation, so Channel Corruption may not be used.