Rulings
Q1: If the played Eosmon is removed from the battle area before this card's All Turns effect activates, can I still delete 1 of my opponent's Digimon with as much or less DP as the played Eosmon?
Yes, you can. The information on Eosmon is referenced at the time it's played, therefore you can perform deletion even if that Eosmon is removed from the battle area before the effect activates.
Q2: My BT6-083 Eosmon was played, then this card's All Turns effect and another effect triggered simultaneously. I activated the other effect, then my BT6-083 Eosmon's DP increased from 4000 to 5000. If I then activate this card's All Turns effect, what is the DP of Digimon I can delete?
5000 DP or less. Unless specified otherwise, the state when the effect is processed will be referenced. In this case, processing was performed when your BT6-083 Eosmon had 5000 DP, therefore you can delete a Digimon with 5000 DP or less.
Q3: My Eosmon was played, then this card's All Turns effect and another effect triggered simultaneously. I activated the other effect, then my Eosmon was removed from the battle area. If I then activate this card's All Turns effect, can I delete an opponent's Digimon?
No, you can't. If the same reference to an "X" trigger condition being performed is made for an effect when it has a "when X" trigger condition, the state when the effect has triggered will be referenced. Because the Digimon that could be referenced by this card's All Turns effect was removed, you can't delete a Digimon.
Data Sources
Lv.6 Eosmon BT17-076 SR Mega | Unknown | Unknown [Digivolve] Lv.5 [Eosmon]: Cost 4 [When Digivolving] [When Attacking] [Once Per Turn] You may play 1 level 5 or lower [Eosmon] from your hand without paying the cost. [All Turns] [Once Per Turn] When one of your [Eosmon] is played, delete 1 of your opponent's Digimon with as much or less DP as that Digimon. [Your Turn] For each Tamer, all of your [Eosmon] get +1000 DP.
Lv.6 Eosmon BT17-076 SR Mega | Unknown | Unknown [Digivolve] Lv.5 [Eosmon]: Cost 4 [When Digivolving] [When Attacking] [Once Per Turn] You may play 1 level 5 or lower [Eosmon] from your hand without paying the cost. [All Turns] [Once Per Turn] When one of your [Eosmon] is played, delete 1 of your opponent's Digimon with as much or less DP as that Digimon. [Your Turn] For each Tamer, all of your [Eosmon] get +1000 DP.
Lv.6 Eosmon BT17-076 SR Mega | Unknown | Unknown [Digivolve] Lv.5 [Eosmon]: Cost 4 [When Digivolving] [When Attacking] [Once Per Turn] You may play 1 level 5 or lower [Eosmon] from your hand without paying the cost. [All Turns] [Once Per Turn] When one of your [Eosmon] is played, delete 1 of your opponent's Digimon with as much or less DP as that Digimon. [Your Turn] For each Tamer, all of your [Eosmon] get +1000 DP.
Lv.6 エオスモン BT17-076 SR 究極体 | 不明 | 不明 Lv.5の「エオスモン」:コスト4 【進化時】【アタック時】[ターンに1回]自分の手札から、Lv.5以下の「エオスモン」1枚をコストを支払わずに登場できる。 【お互いのターン】[ターンに1回]自分の「エオスモン」が登場したとき、そのデジモンのDP以下の相手のデジモン1体を消滅させる。 【自分のターン】テイマー1体ごとに、自分の「エオスモン」全てをDP+1000。