No, mutating a creature does not count as casting a creature spell.
No, copying a spell does not count as casting it.
No, cycling does not count as casting a spell in the game.
No, copying a spell in Magic: The Gathering does not count as casting it.
'Casting a creature spell' means to pay the mana cost and put the creature spell on the stack. If this is allowed to resolve, then the creature enters the Battlefield.
Yes, replicating a spell counts as casting it.
No, casting a creature spell does not allow you to draw a card.
To cast a Magic: The Gathering creature spell, a player must have enough mana of the appropriate color and type required by the spell. The player then announces they are casting the spell, pays the mana cost, and places the creature card onto the battlefield. The creature spell then resolves, entering the battlefield and becoming a permanent creature under the player's control.
Yes, they do count as being permanents, and will be affected by any spell or ability that can affect them.
Hostility has a replacement effect for when your 'spell' deals damage to the opponent. Creatures are indeed summoned by casting a 'Creature Spell' but when this resolves, it is simply now just a 'creature'. Any damage they deal, effect or battle, is not 'spell' damage. I don't know if there are any situations where a Creature Spell can deal damage while it remains a spell. Damage dealt when creature spells are cycled, etc, are damage from an ability, not from a spell.
No, copying a spell is not considered casting it.
Transfiguration is spell casting.
In Magic: The Gathering, the mechanic "convoke" allows you to tap creatures to help pay for the cost of casting a spell. When you use convoke to cast a creature spell, that creature does not have summoning sickness and can attack or use abilities right away.