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: Does this card's Your Turn effect trigger when this card itself is played?

Yes, it triggers.

Q3: My BT6-083 Eosmon was played, then this card's All Turns effect and another effect triggered simultaneously. If another effect activates and the DP of my played BT6-083 Eosmon changes from 4000 to 5000, then I activate this card's All Turns effect, what is the DP of Digimon I can delete?

5000 DP or less. If the same reference to an "X" trigger condition being performed isn't being made for an effect when it has a "when X" trigger condition, the state when the effect has triggered 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.

Q4: Another of my Eosmon was played, then this card's All Turns effect and another effect triggered simultaneously. I activated the other effect, then my played 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 isn't being made for an effect when it has a "when X" trigger condition, the state when the effect has triggered will be referenced. In this case, because the Digimon that could be referenced by this card's All Turns effect was removed, you can't delete a Digimon.

Q5: Does this card's Your Turn effect trigger when this card itself plays?

Yes, triggered.

Q6: My "BT6-083 Eosmon" appeared and this card's All Your Turns effect and another effect triggered simultaneously. If the DP of my "BT6-083 Eosmon" that played after the other effect is activated goes from 4000 to 5000, which DP or less can be deleted by the All Your Turns effect of this card that was triggered after that?

DP5000 or less. If the triggering condition is "when you do 00" and the effect does not have the same reference as the triggering condition of "when you do 00," then it refers to the situation at the time the effect is being processed. In this situation, the DP of your "BT6-083 Eosmon" is 5000 at the time it is being processed, so you can delete Digimon with DP 5000 or less.

Q7: Other your "Eosmon" appeared and this card's All Your Turns effect and other effects triggered at the same time. My "Eosmon" which appeared and activated from the other effect has left the Battle Area. Can I delete my opponent's Digimon with this card's Opponent's Turn effect activated after that?

No, it cannot be deleted. If the triggering condition is "when you do 00" and the effect does not have the same reference as the triggering condition of "when you do 00," then the situation at the time the effect is processed is referenced. In this situation, the card cannot be deleted because there are no more Digimons that can be referenced by the card's All Turns effect.