As mentioned in some comments, this seems to mimic the Bite attack from the vampires.
Bite (Bat or Vampire Form Only): Melee Weapon Attack: +9 to hit, reach 5 ft., one willing creature, or a creature that is grappled by the vampire, incapacitated, or restrained. Hit: 7 (1d6 + 4) piercing damage plus 10 (3d6) necrotic damage. The target's hit point maximum is reduced by an amount equal to the necrotic damage taken, and the vampire regains hit points equal to that amount. The reduction lasts until the target finishes a long rest. The target dies if this effect reduces its hit point maximum to 0. A humanoid slain in this way and then buried in the ground rises the following night as a Vampire Spawn under the vampire's control.
With that in mind, answering your questions:
- No. The text states "Make a melee attack against the target.", this implies this is the action you are taking. It doesn't state "As a part of another melee attack" or "When you make a melee attack". The text is poorly written and inconsistent with other features, but seems clear enough (i.e. not ambiguous).
- I'm not sure about what the rules mean by "special", or how it's relevant. It is a melee attack, though.
- This is where the text is completely unclear. As it doesn't state a melee weapon attack, it is completely up to the DM. It is probably intended that it is a melee weapon attack, similar to how Bite is. It is also usual that creatures have proficiency with their attacks (such as the Bite case), but I'm not certain about race features. For example, the Aarakocra feature explicitly states
Talons. You are proficient with your unarmed strikes,
which deal 1d4 slashing damage on a hit
although this talon attack is intended to replace the unarmed strike.
Anyway, depending on how you rule if the vampire is or not proficient and if it is or not a melee weapon attack, you can calculate the modifier to hit as usual.
The most consistent (with Bite) way to rule it is making it a melee weapon attack and having proficiency with it, which would lead to a modifier of +bonus proficiency +Str.
Finally, note that the Plane Shift: Ixalan article has the disclaimer
The game mechanics in this supplement are usable in your D&D campaign but are not fully tempered by playtests and design iterations. For these reasons, material in this supplement is not legal in D&D Organized Play events.
similar to Unearthed Arcana (though with different purposes - UA is supposed to be playtested by the other tables). This means the writing is indeed inconsistent and most of the ruling and interpretation ends up to DM fiat.
I'm creating this answer because I disagree with the Ambiguous statement and using Physical to refer to the type of attack, when physical is actually used in the books for types of damage (slash./pierc./bludg.), which necrotic is not a part of. Yeah, kinda nitpicking, but anyway.