“The dead magic trait functions in all respects like an antimagic field spell”
This doesn’t cover other forms of Dead Magic Zone, which are not actually addressed specifically (that I can find), but presumably those are also equal to antimagic field.
So in all ways, a Dead Magic Zone is identical to an antimagic field except, most likely, for the duration and area.
Incorporeal Undead have no issue with antimagic field
Nowhere in either the spell’s description or in the description of the Incorporeal subtype are they called out as suppressed by antimagic field. The spell itself says that
Elementals, undead, and outsider are likewise unaffected unless summoned.
Antimagic field “prevents the functioning of any magic items or spells within its confines”
Potions and wands (or dorjes) are magical items, and their effects are suppressed:
An antimagic field suppresses any spell or magical effect used within, brought into, or cast into the area, but does not dispel it. Time spent within an antimagic field counts against the suppressed spell's duration.
Nothing about an antimagic field prevents activating a magic item, the effect will simply be suppressed until not under the field. In fact, nothing here says someone in the field is prevented from casting spells, either: you simply get no effect until you are no longer under the field.
So if you use a wand or dorje, you spend a charge, the spell effect goes into place, but is immediately suppressed. If you drink a potion, it is consumed and gone, with no magic taking place immediately. But in both cases, if you leave the antimagic field, the magic springs to life and the effects function normally for any duration they have remaining (assuming they would have any; instantaneous effects are out of luck).
These effects are poor game design and I recommend against using them.
Antimagic field is a weak spell because it is high-level and has a small area. As a result, its caster is guaranteed to be shafted by it – he’s a high-level mage, after all – and his targets can easily escape it. The ability to cast instantaneous Conjurations into the field only makes it more pitiable.
Dead Magic Zones have the opposite problem. They aren’t cast, their area can be huge, and their duration can be infinite. There’s typically no source so they cannot be turned off by attacking something. That means they’re just pure DM-fiat, and there are no relevant tactical decisions that can do anything about them.
Worse, Dead Magic Zones are highly asymmetric: spellcasters can be left with literally nothing to do, warriors stripped of magic items are in a lot of trouble, and plenty of monsters don’t care in the least. So some people are basically told “you don’t get to play,” others get “here’s a challenge that you have to complete without any of your usual abilities,” and some (most normally, the enemy) are told: “hey, your enemies have been stripped of most of their ability to fight, have fun!”
I therefore suggest not using them at all. If you insist on using them, use them sparingly, and make sure that there are plenty of options for dealing with it. A fight, or series of fights, or a whole dungeon, that takes place entirely within a Dead Magic Zone that has no source and cannot be removed, is boring and poor design. Some characters may literally be relegated to cheerleader status, without even the Bard’s ability to make his cheers count for something (even if he was a Bard, since Inspire Courage is supernatural).
Instead, a fight can be made more interesting by temporary or small Dead Magic Zones, or Dead Magic Zones which can be moved, turned on, or turned off (or which simply do these things automatically, so you have to try to predict/time your positioning to take advantage of them/avoid getting screwed by them). This makes the DMZ add tactical dimensions to the situation, rather than detract from them.
Despite the use of the word “tactical” in these points, this isn’t purely about combat. Combat is a major part of Dungeons & Dragons; the overwhelming majority of the rules are related to it, so it gets proportional coverage here. But plenty of non-combat situations are impossible to deal with without magic, as well.
In short, D&D 3.x is extremely high-magic, and after about level 4 or so the game starts assuming that you have it. By the time antimagic field is available, magic is very nearly everything. Removing magic tends to work poorly and has to be done very carefully.
DMZs also work OK as a way of attempting to enforce non-combat solutions to disagreements with NPCs. They could be a great way to force the PCs to actually talk with someone rather than dash their brains out, if said someone is, say, on the far side of a long chasm with a DMZ between them so they cannot fly over it. This is basically an attempt to intentionally use the fact that DMZs prevent regular play in order to force some non-mechanical roleplaying. In some cases, these may be necessary even among good roleplayers, since it's hard to justify, in-character, not just killing the BBEG when he wants to talk, if you have that opportunity.