Q2: Can I use this card's Security effect to play a Digi-Egg card with the LIBERATOR trait?
No, you can't. You can't play it because Digi-Egg cards don't have play costs.
Lv.4 ShoeShoemon ST19-08 SR Champion | Virus | Puppet/LIBERATOR [Security] You may play 1 card with the [LIBERATOR] trait and a play cost of 4 or less from your hand or trash without paying the cost. <Overclock ([Puppet] Trait)> (At the end of your turn, by deleting 1 of your Tokens or other [Puppet] trait Digimon, this Digimon may attack a player without suspending.) --- Inherited: [Your Turn] All of your opponent's Security Digimon get -3000 DP.
Lv.4 ShoeShoemon ST19-08 SR Champion | Virus | Puppet/LIBERATOR [Security] You may play 1 card with the [LIBERATOR] trait and a play cost of 4 or less from your hand or trash without paying the cost. <Overclock ([Puppet] Trait)> (At the end of your turn, by deleting 1 of your Tokens or other [Puppet] trait Digimon, this Digimon may attack a player without suspending.) --- Inherited: [Your Turn] All of your opponent's Security Digimon get -3000 DP.
Lv.4 ShoeShoemon ST19-08 SR Champion | Virus | Puppet/LIBERATOR [Security] You may play 1 card with the [LIBERATOR] trait and a play cost of 4 or less from your hand or trash without paying the cost. <Overclock ([Puppet] trait)> --- Inherited: [Your Turn] All of your opponent's security Digimon get -3000 DP.
Lv.4 シューシューモン ST19-08 SR 成熟期 | ウィルス種 | パペット型/リベレイター 【セキュリティ】自分の手札/トラッシュから、特徴に「リベレイター」を持つ登場コスト4以下のカード1枚をコストを支払わずに登場できる。 ≪オーバークロック《特徴「パペット型」》≫(自分のターン終了時、自分のトークンか特徴に「パペット型」を持つ他の自分のデジモン1体を消滅させることで、このデジモンでレストせずにプレイヤーにアタックする) --- Inherited: 【自分のターン】相手のセキュリティデジモン全てをDP-3000。