Q1: Can I use this card's End of Your Turn effect to grant "Quick Attack" to one of my Digimon and then not attack with that Digimon without rest?
No, it cannot. Digimon that are given "Quick Attack" by this effect will attack without rest as much as possible.
Q2: When attacking with this card's End of Your Turn effect, can I attack with a Digimon in Rest state?
Yes, you can attack with a Digimon in rest state.
Q3: I have two copies of this card in my Battle Area. At the End of Your Turn, I attacked with the first card's End of Your Turn effect. Can I attack with the second card's End of Your Turn effect after that?
No, you cannot attack with the effect of the second card. While attacking, you cannot declare a new attack. In this situation, at the end of your Turn, the two When End of Your Turn effects of this card are triggered simultaneously, and you attack with the first End of Your Turn effect. Before the Counter timing, the second When End of Your Turn effect can be activated, but since you cannot declare an Attack during the Attack, the second card cannot attack even if you activate it.
Lv.7 Omnimon (X Antibody) BT20-102 SEC Mega | Vaccine | Holy Warrior/X Antibody/Royal Knight/LIBERATOR [Digivolve] [Omnimon]: Cost 2 <Raid> <Piercing> <Blocker> [On Play] [When Digivolving] If [Omnimon] or [X Antibody] is in this Digimon's digivolution cards, choose 1 of both players' Digimon and delete all other Digimon. Then, return 1 of your opponent's Digimon to the bottom of the deck. [End of Your Turn] [Once Per Turn] 1 of your Digimon may gain <Rush> for the turn and attack without suspending.
Lv.7 Omnimon (X Antibody) BT20-102 SEC Mega | Vaccine | Holy Warrior/X Antibody/Royal Knight [Digivolve] [Omnimon]: Cost 2 <Raid> <Piercing> (When this Digimon attacks and deletes an opponent's Digimon, it performs any security checks it normally would.) <Blocker> [On Play] [When Digivolving] If [Omnimon] or [X Antibody] is in this Digimon's digivolution cards, choose 1 of both players' Digimon and delete all other Digimon. Then, return 1 of your opponent's Digimon to the bottom of the deck. [End of Your Turn] [Once Per Turn] 1 of your Digimon may gain <Rush> for the turn and attack without suspending.
Lv.7 オメガモンX抗体 BT20-102 SEC 究極体 | ワクチン種 | 聖騎士型/X抗体/ロイヤルナイツ/リベレイター 「オメガモン」:コスト2 ≪突進≫≪貫通≫≪ブロッカー≫ 【登場時】【進化時】このデジモンの進化元に「オメガモン」/「X抗体」があるなら、お互いのデジモン1体ずつを選び、選んだデジモン以外のデジモン全てを消滅させる。その後、相手のデジモン1体をデッキの下に戻す。 【自分のターン終了時】[ターンに1回]ターン終了まで、自分のデジモン1体は≪速攻≫を得て、そのデジモンでレストせずにアタックできる。