You are never really floating freely, all gravitational wells extend to infinity. The curvature just becomes really small so the resulting acceleration is infinitesimally small too.
There is always microgravity, i.e. residual acceleration due to drag and thrusters, and especially vibrations.
Worst offenders are actually the astronauts, especially when they work out on one of their exercise devices right next to your experiment
Thankfully, the same measurement technique can be used to measure e.g. gravitational fields, and there are already devices available on the market which surpass classical gravimeters in accuracy and precision. There are also satellite missions in the planning to map Earth's gravitational field with atom interferometry.
That's the gist of it, yes. The big advantage atom interferometers have over classical IMUs is that they do not need to be recalibrated every so often. They are thus often called "drift free". That still means that you need to compare your position with a reference after a certain time, but the position uncertainty is much lower and only determined by the error accumulating from integration, not due to the instrument itself degrading over time and introducing systematic errors. The reason here is not really "quantum magic" but rather the fact that the reference for the measurement is a laser beam which can be controlled with remarkable precision and accuracy.
You don't need any helium or nitrogen here, cooling happens only by laser cooling and evaporative cooling from magnetic or optical traps. The atoms are perfectly insulated in an ultra high vacuum.
Electronics still take the bulk of the volume here, as does the laser system. While the lasers themselves are tiny indeed, the light needs to be manipulated before reaching the atoms.
And yes, it involves quite a lot of fiber optics :-).
There is no liquid nitrogen involved here. The instrument from the article is actually rather big, current generations of quantum IMUs are roughly half this size with lots of room for miniaturization.
One big advantage of these atom interferometers is that they actually don't need to be recalibrated because the reference is the wavelength of the lasers which can be controlled with extreme precision.
A big disadvantage is however the limited repetition rate, which is on the order of only 1 Hz at the moment. Currently, combinations with "classical" IMUs seem most promising, and there is lots of interest in these devices for applications in planes, cars and spacecraft.