Rulings
Q1: Can I not attack with this card after this card's On Play When Digivolving effect grants "Penetrating" and DP+4000 to one of my Digimon?
Yes, it is possible.
Q2: With the When Digivolving effect of "BT16-077 Dinobeamon", I played this card On Play and attacked a player with "BT16-077 Dinobeamon" with "Quick Attack" as it is. After that, can I attack with this card by its On Play When Digivolving effect?
No, you cannot attack. While attacking, you cannot declare a new attack. In this situation, you can attack with "BT16-077 Dino Beamon" under the When Digivolving effect of "BT16-077 Dino Beamon". Before the Counter timing, the card's On Play and When Digivolving effects can be activated, but you cannot declare an attack during the attack, so you cannot attack even if you activate it.
Q3: If this card's All Turns effect is activated and two Digimon including the leaving Digimon digivolve, does the digivolved Digimon leave the Battle Area?
No, they do not leave the Battle Area. DNA digivolves digivolve as a new Digimon, so it is a different Digimon than the one that leaves.
Data Sources
Lv.5 Paildramon BT20-016 U Ultimate | Free | Dragonkin [DNA Digivolve] Red Lv.4 + purple Lv.4: Cost 0 [On Play] [When Digivolving] For the turn, 1 of your Digimon gains <Piercing> and gets +4000 DP. Then, this Digimon may attack. [All Turns] When any of your [Paildramon] or [Dinobeemon] would be deleted, 2 of your Digimon may DNA digivolve into [Imperialdramon: Dragon Mode] in the hand. --- Inherited: <Security A. +1> (This Digimon checks 1 additional security card.)
Lv.5 Paildramon BT20-016 U Ultimate | Free | Dragonkin [DNA Digivolve] Red Lv.4 + Purple Lv.4: Cost 0 [On Play] [When Digivolving] For the turn, 1 of your Digimon gains <Piercing> and gets +4000 DP. Then, this Digimon may attack. [All Turns] When any of your [Paildramon] or [Dinobeemon] would be deleted, 2 of your Digimon may DNA digivolve into [Imperialdramon: Dragon Mode] in the hand. --- Inherited: <Security A. +1>.
Lv.5 パイルドラモン BT20-016 U 完全体 | フリー | 竜人型 〔ジョグレス〕赤Lv.4+紫Lv.4:コスト0 指定のデジモン2体を重ね、アクティブで進化する 【登場時】【進化時】ターン終了まで、自分のデジモン1体は≪貫通≫を得て、DP+4000。その後、このデジモンでアタックできる。 【お互いのターン】自分の「パイルドラモン」/「ディノビーモン」が消滅するとき、自分のデジモン2体で手札の「インペリアルドラモン：ドラゴンモード」にジョグレス進化できる。 --- Inherited: ≪Sアタック+1≫（このデジモンがチェックする枚数+1）