At the start of the level, the game goes through each Thing in the map in sequence and spawns the appropriate object. If that Thing is a player start, the game will spawn a player object if that player is in the game (player 1 is always present in single-player games, players 2-4 are only spawned in multiplayer, obviously), and associate the newly spawned object with that player (for viewpoint, movement, etc). If there is more than one player start for a particular player in single player or coop (deathmatch starts work differently, and voodoo dolls cannot be spawned in deathmatch), the game will simply spawn a duplicate player object, and associate the player with the new player object, leaving multiple player objects associated with the player. Only the most recently spawned player object (the highest-numbered Thing) will have the player's viewpoint and movement (the others just stand still and you can't see their point of view), but all of the player objects have their damage associated with the player's health. Thus damaging either player object will cause the player's health to decrease.
OK, so is there a definite explanation of what are voodoo dolls, what can they do, how they work?
The zombie player glitch results because health is not the same thing as hit points. All Things have hit points (but only shootable Things actually use them), including the player object(s), but the player's health (as indicated on the status bar) is not the same thing. Normally this is never noticed, because damaging a player object causes the player's health to be reduced by the same amount, so the numbers should always be identical. When voodoo dolls are involved, they are not. For example, if the player and a voodoo doll are both hit for 10 damage, the player will lose 20 health, but the player object will only lose 10 hit points. Thus the health can be desynced from hit points, which leads to a problem when one of the two numbers drops to zero.
Doom considers a monster or player to be dead when the Thing's hit points are reduced to zero. This has the consequence that reducing a voodoo doll's hit points to zero will cause the player to "die" (faceplant, can't move, can't do anything except restart/respawn), regardless of how much health he actually has. However, the real player object still has nonzero hitpoints, so it can still be attacked, and may eventually "die" again. In fact, if you respawn in multiplayer, you can see your "corpse" still standing there, and you can "re-kill" it yourself. This will not harm you, since respawning disassociates the player from the "corpse" (but not from any surviving voodoo dolls, so they will still be active).
The case 1 zombie is the reverse of this. The player's health is reduced to zero (triggering the loss of his weapon), but not the player object or the voodoo doll's hit points. With nonzero hit points, the player is not considered to be dead, and can move around and do most things that a living player can do (except fire or wake up monsters, since monsters can't be awoken by dead players, and the player's health is used to determine this).
Exiting the level causes a new player object to spawned, with its hit points set to the player's current health, ending all zombie shenanigans... unless the player still had zero health when exiting the level. There is special handling for dead players exiting levels, but since the player isn't considered dead, this doesn't come into play. Instead, the new player object is spawned with zero hit points, and is now considered truly dead (cannot be attacked, cannot pick up items, etc), except that since it didn't go through the normal death sequence, the view height is still normal, and you cannot restart/respawn.
That's how zombie players work. The other main effect of voodoo dolls is that they can, if propelled by an outside force (directional damage or conveyor belt), pick up items and trigger walkover linedefs (including level exits), and the game will treat these events as though they were triggered by the corresponding player (items picked up by a voodoo doll will be added to the player's inventory, for instance).
In all other respects, voodoo dolls act like monsters (or rather Keens, since they cannot voluntarily move or attack). Voodoo dolls do not wake up monsters, and monsters will never deliberately attack a voodoo doll (unless provoked, but that's impossible because voodoo dolls cannot attack). Voodoo dolls can be damaged by crushing ceilings but not damaging floors, and cannot telefrag anything except on MAP30 (though the real player can always telefrag any voodoo doll, even his own, which is used in some instant death traps). Dead voodoo dolls cannot be resurrected by Arch-Viles (because they have no resurrection frame), do not respawn in Nightmare! mode (because they don't count as kills), nor do they respawn in multiplayer when the corresponding real player does (which means any tricks involving the death of a voodoo doll will only work once in multiplayer).
I think that covers just about everything. Note that this only applies to vanilla Doom, Boom, and strictly compatible ports. Other source ports may alter voodoo doll behaviour (in particular, Doom Legacy has some neat enhancements for voodoo doll behaviour, but that's a whole new topic).