Obviously - beause otherwise the ability/bonus woul be nonsense - all "cancel X" go before everything else. And there is always one solution to logically solve all cancels in any situation. In above situation, all cancels apply but your cancel cancels your opponent's cancel, leaving your other aboloty intact. I mean, how else shall it work without ignoring ability/bonus?
They are both put on the table.
Both, the Bonus, and the ability are put forth for both cards in each round.
if one card has a stop ability, or bonus, then you remove the opposing ability or bonus, depending.
then the battle takes place.
now, when you focus in on cards with the same ability as a bonus, or similar, then you find it slightly harder to explain.
a card with - opp 12 attack min 8 bonus, and Ability -5 opp attack min 2
then you use both together, the one with the higher min acts first, but the total min does not go down.
so if the opponent card has 8 attack, or more, the -12 brings it to 8, then the -5 brings it to 3, but if they had less than 8, then the -5 brings them to 2.
does that make any sense?
If two abilities in the slots in which they would cancel each other go up against each other, they effectlvely both work. Not that it really matters since they're conceliing each other out and effectively do nothing anyway.
(Ie if a Cancel Bonus in the Ability slot runs into a cancel Ability in the Bonus slot, then they basically both work and the abilities negate one another).
If one is not canceled (Ex: You have Cancel Bonus in the Bonus slot, and Cancel Ability in the Abilitty Slot, and they have Cancel Ability in the Bonus slot, their Cancel Ability would not function as your Cancel Bonus would stop it. Then your Cancel Ability would still work and stop the opponent's Ability).
This question has been answered many times before. There is no "First" in this game. All abilities and bonues are applied simultaneously, and then checked to see if any of them are canceled.