You wouldn't be able to direct all of it off-axis. In order to do any meaningful work with it, some of it must be lost as heat or the entropy of the universe does not increase. That would violate the laws of thermodynamics. Essentially, you can reduce the entropy of an isolated collection of particles, but only at the expense of increasing the entropy of another collection of atoms, and only if you move the entropy around.
There's a slim possibility that a sufficiently advanced civilization has improved the efficiency of their process sufficiently to the point where the thermal radiation isn't measurable with our technology, but there still must be thermal radiation if any meaningful work is being done with the harvested energy.
Yes, but not necessarily thermal radiation aimed in the narrow direction necessary for us to detect it. Thermodynamics does not guarantee that all necessary information propagate in nice measurable spheres for all to see equally.
There's a slim possibility that a sufficiently advanced civilization has improved the efficiency of their process sufficiently to the point where the thermal radiation isn't measurable with our technology, but there still must be thermal radiation if any meaningful work is being done with the harvested energy.