As Stronghold Arena's last ability resolves, you don't get to see what the card of your library is before deciding whether to reveal it. If you choose to reveal it, you must put it into your hand and will lose life.

You may kick Stronghold Arena only once for its {G} cost and only once for its {W} cost. You can kick it twice by paying {G}{W}, but you can't kick it twice by paying {G}{G} or {W}{W}. You can't kick it more than twice.

Normally, combat damage is dealt all at the same time. In that case, Stronghold Arena's last ability triggers once for each opponent that creatures you control dealt combat damage to, regardless of how many creatures were dealing that damage. If any of those creatures have double strike or first strike, the ability will trigger once for each opponent dealt damage in each combat damage step.

If you put a permanent with a kicker ability onto the battlefield without casting it, you can't kick it.

To determine a spell's total cost, start with the mana cost (or an alternative cost if another card's effect allows you to pay one instead), add any cost increases (such as kicker), then apply any cost reductions. The spell's mana value remains unchanged, no matter what the total cost to cast it was.

If a card or token enters as a copy of a permanent, the new permanent isn't kicked, even if the original was.

If you copy a kicked spell on the stack, the copy is also kicked. If the copied spell is a permanent spell, the token the copy of that spell becomes when it enters is also kicked.

If a spell's kicker cost was paid, the spell is "kicked."

The kicker ability doesn't let you pay a kicker cost more than once.