Regarding your original question: “A question I've been wondering about is whether it is possible to fa jing without being rooted. For example, if one were bear hugged and lifted into the air, is fa jing still usable to break the hold?”
It’s difficult to address this hypothetical. Part of the difficulty lies in establishing what fajin would mean in this context. There are a number of things one could do to break the hold. Whether those things would fit a classical taijiquan definition of fajin is another matter. Another part of the difficulty lies in what “being rooted” means. Just for the sake of argument, one could make a case that if I were being bear hugged and lifted off the ground my structure and alignment would be compromised. However, there is still a root; I’m connected to the ground through the person holding me. It would be better to avoid getting into this situation, but I’m connected to the ground in any case.
Not much is said about root in classical taiji documents that I can recall. The Taijiquan Classic, sometimes attributed to Wu Yuxiang, mentions “root” (gen) exactly twice. The first is in the sentence, “It is rooted (gen) in the feet, issued by the legs, governed by the waist, and expressed in the fingers.” The second is in reference to lifting “an object,” where it is stated that one must first apply a “dampening force” (cuo). “Thus, its root (gen) will be severed, and it will be collapsed quickly and decisively.” Presumably, the “object” is one’s opponent, and the wording actually implies that the opponent severs his own root (qi gen zi duan) by losing his balance.
In my opinion, both of these mentions of “root” are relevant to what might be called the most optimal context for fajin. That is, to use your words, it is a context that involves “proper body alignment to create a stable connection with the earth,” described in the taiji classic as a progression rooted in the feet and up through the torso into the fingers. Also, in this optimal context, one applies fajin against an opponent whose root has already been broken. The breaking of the opponent’s root is otherwise expressed in the line from the Song of Pushing Hands, “Attract him into emptiness, join, then issue.”