Game lore explanations
The lore of the game attributes to dark energy to both attractive and repulsive effects. And it says that the "element zero" is able to manipulate the mass of the objects (instead of manipulating gravitational fields as I described below). If so, it could be better understood as manipulating higg bosons. I find this explanation problematic because it seem to suggest negative mass for repulsion.
I found this description of the "element zero":
When subjected to an electrical current, the rare material dubbed element zero, or "eezo", emits a dark energy field that raises or lowers the mass of all objects within it. This "mass effect" is used in countless ways, from generating artificial gravity to manufacturing high-strength construction materials. It is most prominently used to enable faster-than-light space travel.
There are two relevant bits to take from here:
it is not said to be a chemical element, but a material. It could be a particular isotope, or it could be a compound, or it could not even be baryonic matter.
The quote says When subjected to an electrical current, that is, they are making a "wire" of the stuff. So, there is no reason to think that the material in it natural state has any noticeable gravitational effects.
The provided link about Drive Charge says:
Element zero FTL drives accumulate a static electrical charge when a vessel has been in FTL flight for some time. This charge steadily increases with the amount of time a vessel spends in FTL. Eventually, it must be discharged.
According to that, an electric charge is accumulated while traveling FTL. It doesn't seem to be an artifact of the FTL drive, but of the FTL travel itself.
Your question
Are element zero's properties internally consistent? Would it indeed allow the applications purported?
No, it doesn't work.
If we consider that "element zero" may also increase mass, then it could explain giving a strong gravitational field to an object. Reducing mass alone will never explain artificial gravity.
Increasing and decreasing mass of objects makes fine control harder, for instance, you could explain deflecting projectiles by increasing the mass of another object that then attracts the projectiles. But doing that would also affect other things in the surrounding.
Similarly, giving mass to an object in front of you so that you "fall" to that object won't allow you to go pass that object. Then you need to reduce the mass of that object again, and increase the mass of another object in the direction you want to go.
Also consider that if you reduce the mass an object to zero it will move at light speed. Alternatively you could understand not as reducing mass, but as reducing the influence of gravity, then you get weightlessness.
This doesn’t really explain repulsion of FTL travel either. Building an Alcubierre drive would still have the placement of matter difficulty.
I will not go into the problem of the electric charge from FTL travel here, because I'm saying that the game lore explanation is not good enough for FTL travel.
A better explanation
What this "element zero" seems to be doing is manipulating the gravitational field, to create either repulsive or attractive gravitational charges. So, it is making dark matter (attractive, but otherwise undetectable) and dark energy (the repulsive equivalent).
Using "element zero" would require some form of energy input. To remain consistent we would say that it uses electricity, and the output is an artificial gravitational charge (positive to be attractive and negative to be repulsive). The machine must be built in such way that allows directing such gravitational charge to the desired location.
The problem is: what happens when a charged particle arrives to an object made of "element zero"? We need to be able to make it so it can modify the surrounding gravitational field to create repulsive and attractive forces. So, there must be two ways to interact with it.
Given that the "element zero" is not a chemical element, I don't see any problem saying it is an anisotropic material that will react differently depending on the direction on which a current traverses it. That could explain how to create an attractive force by using current in a set of directions, a negative force in a different set of directions, and none at all in yet another set of directions.
Under this idea, you need use a different current direction to have a different effect, and you would need to actually rotate the system to direct it. The intensity of the effect would be linked to the power used.
Your question
Are element zero's properties internally consistent? Would it indeed allow the applications purported?
Yes.
By creating artificial gravitational fields (both attractive and repulsive) relative to the machine that generates them can be used to move objects at distance. And it can be used to negate gravitational attraction, including those created using it.
Under the idea that it manipulates the gravitational fields relative to itself, it would be a way to create an Alcubierre drive. So it can explain FTL travel.
To explain an electric charge built up in an object traveling FTL using this method, such that it requires discharging it... I would have to say that the object captures charged particles during the trip※. A longer trip would lead to more charged particles being captured.
※Speculation: We could say that during the trip the ship pushes with it astrophysical plasma, just by it not having time to flow around the ship.
As a consequence of that electric charge, the ships would need a stronger insulation if they are meant to sustain larger FTL trips - as to prevent sparks from the outside to the interior of the ship that may cause damage. Also, leaving the ship after the travel won't be safe unless that charge is canceled.
