The effect "During either player's turn, as Chain Link 3 or higher: You can Special Summon this card from your hand." is a Quick Effect that is activated from the hand. (It cannot be activated as Chain 1 or 2. It can be activated as Chain 3 or higher. It cannot be activated during the Damage Step.)
As Chain Link 3 or higher, you can activate the effect of "Winged Kuriboh LV9" in a Chain, and Special Summon it when its effect resolves. You can only Special Summon 1 "Winged Kuriboh LV9". Even if you have multiple copies of "Winged Kuriboh LV9" in your hand, you can only activate the effect of 1 of them in the same Chain.
After a Spell Card has been activated, if it is sent to the Graveyard as a cost or destroyed by another card effect, it will be banished by "Winged Kuriboh LV9".
LV Monster Rulings
Previously Official Rulings
These TCG rulings were issued by Upper Deck Entertainment and have since been deemed unofficial by Konami. They were previously considered official, but may or may not necessarily hold true nowadays.
If a level monster fulfills its condition to Special Summon the next-highest level (for example, if "Armed Dragon LV5" destroys a monster as a result of battle), and the level monster is then flipped face-down, or removed from play permanently, or sent to the Graveyard and then Special Summoned from the Graveyard, then its condition is reset and it is no longer considered to have met the condition to level up to the next level.