Rulings
Q1: When an effect plays an opponent's Digimon into the breeding area, does this card's All Turns effect activate?
No, it doesn't activate.
Q2: Does this card's All Turns effect also activate when one of my effects plays an opponent's Digimon/Tamer?
Yes, it activates.
Q3: Can I use this card's When Digivolving effect to delete just 1 of my opponent's Digimon if my opponent has level 3, level 5, and level 7 Digimon?
No, you can't. You must delete as many as possible.
Q5: My EX5-063 Leviamon with Security Attack +1 attacked a player, and a Digimon or Tamer was played by a Security effect upon the 2nd security check. This caused the EX5-063 Leviamon to digivolve into this card from the trash. This card has Security Attack +2, but does it get 2 additional security checks?
No, it only gets 1 additional security check. EX5-063 Leviamon's Security Attack +1 is lost upon the digivolution but it gains Security Attack +2, so the total possible security checks will be 3. In the situation from your question, you have already performed 2 security checks, so you have 1 security check remaining.
Q6: I have this card and Leviamon in my trash, then my {{ToCard|EX5-069|Biting Crush was placed in the battle area. If my opponent's Digimon is played by an effect at this time, this card and EX5-069 Biting Crush will trigger simultaneously, but can I activate EX5-069 Biting Crush's effect first to play [Leviamon}} from my trash and then use the effect to digivolve that Leviamon into this card?
Yes, you can.
Data Sources
Lv.6 Leviamon (X Antibody) BT15-081 SR Mega | Virus | Demon Lord/Seven Great Demon Lords/X Antibody [Digivolve] [Leviamon]: Cost 2 [Trash] [All Turns] When an effect plays an opponent's Digimon or Tamer, 1 of your [Leviamon] or Digimon with [X Antibody] in their digivolution cards may digivolve into this card without paying the cost. <Security A. +2> [When Digivolving] If your opponent has as many or more total Digimon and Tamers as you, delete 1 of their Tamers. Then, delete 1 of your opponent's level 3, 1 of their level 5 and 1 of their level 7 Digimon.
Lv.6 Leviamon (X Antibody) BT15-081 SR Mega | Virus | Demon Lord/Seven Great Demon Lords/X Antibody [Digivolve] [Leviamon]: Cost 2 [Trash] [All Turns] When an effect plays an opponent's Digimon or Tamer, 1 of your [Leviamon] or Digimon with [X Antibody] in their digivolution cards may digivolve into this card without paying the cost. <Security A. +2> [When Digivolving] If your opponent has as many or more total Digimon and Tamers as you, delete 1 of their Tamers. Then, delete 1 of your opponent's level 3, 1 of their level 5 and 1 of their level 7 Digimon.
Lv.6 Leviamon (X Antibody) BT15-081 SR
Mega | Virus | Demon Lord/Seven Great Demon Lords/X Antibody
{{DigivolveFromNames|2|Leviamon}}
[Trash] [All Turns] When an effect plays an opponent's Digimon or Tamer, 1 of your [Leviamon] or Digimon with [X Antibody] in their digivolution cards may digivolve into this card without paying the cost.
<Security A. +2>
[When Digivolving] If your opponent has as many or more total Digimon and Tamers as you, delete 1 of their Tamers. Then, delete 1 of your opponent's level 3, 1 of their level 5 and 1 of their level 7 Digimon.Lv.6 リヴァイアモンX抗体 BT15-081 SR 究極体 | ウィルス種 | 魔王型/七大魔王/X抗体 ［トラッシュ］【お互いのターン】効果で相手のデジモン/テイマーが登場したとき、自分の「リヴァイアモン」か進化元に「X抗体」がある自分のデジモン1体をこのカードにコストを支払わずに進化できる。 ≪Sアタック+2≫ 【進化時】相手のデジモンとテイマーの合計体数が自分のデジモンとテイマーの合計体数以上なら、相手のテイマー1体を消滅させる。その後、Lv.3とLv.5とLv.7の相手のデジモン1体ずつを消滅させる。