Q1: Does this card's Your Turn effect give me 1 memory when one of my Digimon moves from the breeding area to the battle area, even if I don't hatch in my breeding area?
Yes, it does. You gain 1 memory when one of your Digimon moves from the breeding area to the battle area, regardless of whether or not you hatch in your breeding area.
Q2: When one of my Digimon moves from the breeding area to the battle area during the breeding phase, does the hatching in the breeding area and gaining 1 memory for this card's Your Turn effect occur during the breeding phase or main phase?
It occurs during the breeding phase. If effects trigger during the breeding phase, the main phase will come after all effects have activated.
Q3: I moved a Digimon from the breeding area to the battle area when this card was in my battle area and all of my Digimon's DP was reduced by activation of an effect. If that Digimon's DP becomes 0 due to the DP-reducing effect at the same time it's moved to the battle area, then it's deleted, does this card's Your Turn effect activate?
Yes, it activates.
Q4: Does this card's Your Turn effect activate when this Digimon itself moves from the breeding area to the battle area?
Yes, it activates.
Lv.3 Ukkomon P-123 P Rookie | Free | Ancient Fairy [Your Turn] [Once Per Turn] When one of your Digimon moves from the breeding area to the battle area, you may hatch in your breeding area. Then, gain 1 memory.
Lv.3 Ukkomon P-123 P Rookie | Free | Ancient Fairy [Your Turn] [Once Per Turn] When one of your Digimon moves from the breeding area to the battle area, you may hatch in your breeding area. Then, gain 1 memory.
Lv.3 Ukkomon P-123 P Rookie | Free | Ancient Fairy [Your Turn] [Once Per Turn] When one of your Digimon moves from the breeding area to the battle area. You may hatch in your breeding area. Then, gain 1 memory.
Lv.3 ウッコモン P-123 P 成長期 | フリー | 古代妖精型 【自分のターン】[ターンに1回]自分のデジモンが育成エリアからバトルエリアに移動したとき、自分の育成エリアに孵化できる。その後、メモリー+1。