Q1: Can this card be placed under a tamer in a "Material Save" of a Digimon with "Shoutmon" in the DigiXros conditions, such as "BT10-013 Shoutmon X5"?
No, it cannot. This card can also be treated as a "Shoutmon" only in DigiXros.
Lv.5 OmniShoutmon BT19-012 R Ultimate | Data | Dragonkin/Xros Heart [Digivolve] [Shoutmon]: Cost 4/Lv.4 w/[Xros Heart] trait: Cost 3 This card is also treated as [Shoutmon] for DigiXros. [On Play] [When Digivolving] 1 of your opponent's Digimon gets -3000 DP for the turn. Then, delete 1 of their Digimon with 3000 DP or less. [On Deletion] You may place 1 Digimon card with the [Xros Heart] or [Blue Flare] trait from your hand or trash under any of your Tamers. --- Inherited: [Your Turn] This Digimon with the [Xros Heart] trait gains <Rush>.
Lv.5 OmniShoutmon BT19-012 R Ultimate | Data | Dragonkin/Xros Heart [Digivolve] [Shoutmon]: Cost 4/Lv.4 w/[Xros Heart] trait: Cost 3 This card is also treated as [Shoutmon] for a DigiXros. [On Play] [When Digivolving] 1 of your opponent's Digimon gets -3000 DP for the turn. Then, delete 1 of your opponent's Digimon with 3000 DP or less. [On Deletion] Place 1 Digimon card with the [Xros Heart] or [Blue Flare] trait from your hand or trash under your Tamers. --- Inherited: [Your Turn] This Digimon with the [Xros Heart] trait gains <Rush>.
Lv.5 オメガシャウトモン BT19-012 R 完全体 | データ種 | 竜人型/クロスハート 「シャウトモン」:コスト4/特徴に「クロスハート」を持つLv.4:コスト3 このカードはデジクロスで「シャウトモン」としても扱う。 【登場時】【進化時】ターン終了まで、相手のデジモン1体をDP-3000。その後、DP3000以下の相手のデジモン1体を消滅させる。 【消滅時】自分の手札/トラッシュから、特徴に「クロスハート」/「ブルーフレア」を持つデジモンカード1枚を自分のテイマーの下に置ける。 --- Inherited: 【自分のターン】特徴に「クロスハート」を持つこのデジモンは≪速攻≫を得る。