Q1: Does this card's Your Turn effect trigger when this card is in the breeding area and would digivolve into a green Digimon card with the TS trait?

No, it doesn't trigger.

Q2: When does "when this Digimon wins a battle" trigger?

It triggers when the battle is won. When the Digimon with this effect wins a battle, the opponent's Digimon that lost the battle is deleted, then the "when this Digimon wins a battle" effect can be activated.

Q3: Does a "when this Digimon wins a battle" effect also trigger when a battle against a Security Digimon is won?

Yes, it triggers.

Q4: A Digimon with a "when this Digimon wins a battle" effect won a battle, and the opponent's Digimon that lost the battle is deleted. At such times, in what order can players activate the "when this Digimon wins a battle" effect and the effects that trigger upon the losing Digimon being deleted?

They trigger simultaneously, so the turn player can activate their effects first.

Q5: A Digimon with a "when this Digimon wins a battle" effect won a battle, and the opponent's Digimon lost the battle. At such times, in what order can players activate the "when this Digimon wins a battle" effect and the loser Digimon's effects such as "when this Digimon would be deleted" and "when this Digimon would leave the battle area" effects?

The "when this Digimon would be deleted" and "when this Digimon would leave the battle area" effects can be activated first.

Q6: A Digimon with a "when this Digimon wins a battle" effect won a battle, and the opponent's Digimon lost the battle. At such times, can the "when this Digimon wins a battle" effect be activated even if an effect prevents the opponent's Digimon from being deleted?

Yes, it can be activated.