Beelzemon

Q:What does this card's "when [...] would leave the battle area other than by battle" mean, exactly?

A:"When [...] would leave the battle area other than by battle" refers to when this Digimon is placed in the trash other than as a result of a battle, returned to the hand/deck, placed in the security stack, moved to the breeding area, or placed under another card.

Q:Can I use this card's [All Turns] effect to place this Digimon itself as the bottom digivolution card of my [Gate of Deadly Sins] in the breeding area?

A:No, you can't. This Digimon hasn't yet been sent to the trash because this effect activates when this Digimon would leave the battle area, so you can't choose it from the trash as the card to place as the bottom digivolution card.