Okay, here's my formula for door vs. Dr.
door can't cast
Elixir of Immortality (EI) on the same turn as another spell without using the upkeep trick. Doing so would allow Dr to
Necrotic Sliver (NS) it, thus ending Dr's ability to cycle. The game really comes down to sequencing
Meddling Mage (MM) on the right card from door's hand. IF MM ever names anything other than EI, door can obtain a draw by gaining 5 life each turn for the rest of the game. Let's put Dr. on the play since door's deck is reactive and can basically play the same from either end.
Dr (1): Play
Volrath's Stronghold (VS), cast MM naming
Bant Charm (BC). (Casting NS doesn't have an immediate impact on the board, especially since door can't cast EI and another spell. If MM named EI, it'd still get charmed one way or another. If it named
Mardu Charm (MC), door would get to EOT BC, upkeep MC for tokens.) End of turn, door casts MC on MM.
door (1): Cast EI. EOT, Dr activates VS to top MM.
Dr (2): Draw MM, cast MM naming BC. EOT, door activates EI. (20/25) Cards get stacked top to bottom EI then MC.
door (2): Draws EI and plays it.
Dr (3): Casts NS. Attacks with MM. (20/23)
door (3): Upkeep BC MM. Mana refills. Draws and casts MC.
Dr (4): Draws MM and casts it naming BC. EOT, door activates EI. (20/28) Cards get stacked top to bottom EI then BC then MC.
door (4): Casts EI. EOT, Dr activates VS to top NS.
Dr (5): Casts NC, attacks with MM. (20/26)
door (5): Draws BC.
Dr (6): Attacks with MM and NS. (20/22)
door (6): Draws MC.
Dr (7): Attacks with MM and NS. (20/18)
door (7): upkeep cast MC on MM, activate EI (20/23). Cards get stacked EI then MC. Draw EI, cast BC on NS. EOT, VS on MM.
Dr (8): Draw and cast MM naming MC.
door (8): Draw MC, cast EI.
Dr (9): Draw and cast NS, attack with MM. (20/21) EOT, activate EI (20/26). Cards get stacked EI then BC.
door (9): Draw and cast EI.
Dr (10): Attack (20/22)
door (10): Draw BC.
Dr (11): Attack, BC the MM. (20/20) EOT, activate EI. (20/25)
So apparently I didn't get to a conclusion where it cycles on that, but I can't really see where a breakthrough is made on either end, so I'm calling it a 2-2.