Tapping a creature with summoning sickness in an equivalent play situation
The answers to #1 and #2 will depend on the judge.  A judge may not give a warning at all if it's unclear any rule has been violated.  But if a judge does give a warning in this situation, it would probably be to you.  The reason is that you are responsible for maintaining a clear and legal game state (both players are responsible for this, but yours are the cards in question).  In the situation you describe, you can't remember where you put the newly summoned creature, which sounds like a failure to maintain a clear game state on your part.  Even if your opponent is mistaken about which one has summoning sickness, the statement "I don't remember" is insufficient defense on your part because the burden is on you to maintain the state of your permanents.

As to #3, you would, in this situation, be able to activate the other Greenseeker.  The reason is that the resolution to an illegal action is, whenever possible, to just rewind to before the illegal action.  In this case, if we undo the activation of one Greenseeker, the other one can be activated.  In the case of the linked question, the only illegal action was attacking with the Mountain, which is undone.  Targeting a Mountain with summoning sickness with Koth was still legal, and so that part of the play stands.