Q1: This card’s effect reads, “While this Digimon has OmniShoutmon or X Antibody in its digivolution cards”. Does having a card with X Antibody in its traits count?
No. This effect specifies a card name, so you must have a card whose name is either OmniShoutmon or X Antibody in this Digimon’s digivolution cards. Having a card with X Antibody in its traits doesn't count.
Lv.5 OmniShoutmon (X Antibody) BT9-013 U Ultimate | Vaccine | Dragonkin/X Antibody Digivolve: 0 from [OmniShoutmon] [When Digivolving] <Blitz> (This Digimon can attack when your opponent has 1 or more memory.)[Your Turn] While this Digimon has [OmniShoutmon] or [X Antibody] in its digivolution cards, this Digimon can also attack your opponent's unsuspended Digimon.
Lv.5 OmniShoutmon (X Antibody) BT9-013 U Ultimate | Vaccine | Dragonkin/X Antibody Digivolve: 0 from [OmniShoutmon] [When Digivolving] <Blitz> (This Digimon can attack when your opponent has 1 or more memory.)[Your Turn] While this Digimon has [OmniShoutmon] or [X Antibody] in its digivolution cards, this Digimon can also attack your opponent's unsuspended Digimon.
Lv.5 OmniShoutmon (X Antibody) BT9-013 U ultimate | Vaccine | Dragonkin/X Antibody [Digivolve] [OmniShoutmon]: Cost 0 [When Digivolving] <Blitz>. [Your Turn] While this Digimon has [OmniShoutmon], [X Antibody] or [link = X Antibody (Card Name)] in its digivolution cards, this Digimon can also attack your opponent's unsuspended Digimon.
Lv.5 オメガシャウトモンX抗体 BT9-013 U 完全体 | ワクチン種 | 竜人型/X抗体 「オメガシャウトモン」から0【進化時】≪進撃≫(メモリーが相手側の1以上のとき、このデジモンはアタックできる)【自分のターン】このデジモンの進化元に、「オメガシャウトモン」か「X抗体」がある間、このデジモンは、アクティブ状態の相手のデジモンにもアタックできる。