Is the EmDrive, or "Relativity Drive" possible?
It is impossible to generate momentum in a closed object without emitting something, so the drive is either not generating thrust, or throwing something backwards. There is no doubt about this.
Assuming that the thrust measurement is accurate, that something could be radiation. This explanation is exceedingly unlikely, since to get mN of radiation pressure you need an enormous amount of energy, since in 1s you get 1 ${\rm gm s^{-1}}$ of momentum, which in radiation can only be carried by $3 \times 10^5$ J (multiply by c), so you need 30,000 Watts of energy to push with mN force, or at least a million Watts for 80 mN. So, it's not radiation.
But a leaky microwave cavity can heat the water-vapor in the air around the object, and the heat can lead to a current of air away from the object. With a air current, you can produce mN thrusts from a relatively small amount of energy, and with a barely noticible breeze. To get mN force, you need to accelerate $300 \ {\rm cm^3}$ of air (1 gram) to 1 m/s every second, or to get 80 mN, accelerate $1 {\rm m^3}$ of air (3000 g) to 0.2 m/s (barely perceptible) and this can be done with a hot-cold thermal gradient behind the device which is hard to notice. If the thrust measurements are not in error, this is the certain cause.
So at best, Shawyer has invented a very inefficient and expensive fan.
EDIT: The initial tests were at atmospheric pressure. To test the fan hypothesis, an easy way is to vary the pressure, another easy way is to put dust in the air to see the air-currents. The experimenters didn't do any of this (or at least didn't publish it if they did), instead, they ran the device inside a vacuum chamber but at ambient pressure after putting it through a vacuum cycle to simulate space. This is not a vacuum test, but it can mislead one on a first read.
In response to criticism of this faux-vacuum test, they did a second test in a real vacuum. This time, they used a torsion pendulum to find a teeny-tiny thrust of no relation to the first purported thrust. The second run in vacuum has completely different effects, possibly due to interactions between charge building up on the device and metallic components of the torsion pendulum, possibly due to deliberate misreporting by these folks, who didn't bother to explain what was going on in the first experiments they hyped up. Since they didn't bother to do a any systematic analysis of the effect on the first run, to vary air-pressure, look at air flows with dust, whatever, or if they did this they didn't bother to admit their initial error, this is not particularly honest experimental work, and there's not much point in talking about it any more. These folks are simply wasting people's time.
Shawyer's "analysis" is a mess, incoherent and deeply confused about fundamental aspects of relativity: he mixes up frames, assumes a universal rest frame, etc. The EmDrive supposedly works best when "stationary relative to the thrust", whatever that means, and Shawyer goes on to suggest using it for levitating vehicles with some kind of conventional propulsion for driving them forward: he apparently believes there is something special about gravitational acceleration.
According to his latest paper, the EmDrive supposedly acts as an electric motor, consuming energy when accelerating and producing it when decelerating. However, a deceleration is just an acceleration in a particular direction, so if it worked, the EmDrive could operate as an infinite energy machine just sitting on one end in a gravity field or while producing thrust for a spacecraft.
So to answer the question in the title: "No." As for other explanations of the observed propulsion, there aren't many details of the measurement procedures or results. There are videos of an EmDrive test on a rotating platform, but there's numerous pieces of equipment that may contain fans, thick power cables going to the equipment that may apply torques, and even a laptop with a hard drive that may be spinning up or down. (And on top of everything else, the whole thing's apparently rotating in the wrong direction.) If this rig is typical of his testing methodology, it's probably safe to chalk up the rest to bad measurements.
No. In special relativity, 4 momentum is exactly conserved. The first component of 4 momentum is total mass/energy, but the next 3 are given by:
p = m*γ(v)*v
m is the invariant mass, how much inertia it has when you are moving at the same velocity of it.
This is Newton except now momentum is a non-linear function of velocity. Nonlinearity does not change anything. Mass and momentum still are constant (ignoring leaks), making γ(v)*v, and thus the center-of-mass velocity v, constant.
So why do we measure force? Possibly currents in the waveguide walls induce currents in the metal support structure which creates small magnetic forces between them.