ooooh, I want to talk about the first part of this post, the bit about Ichigo's sword going wobbly! This is another one of those incredibly interesting things that happens in early Bleach that's easy to forget. In this case, though, I think it does connect to lore we get later on in the series.
This is just my theory, but...
I think this is not a unique Ichigo thing, but it is a result of him being thrown into the world as a shinigami without knowing what he's doing. I think, very specifically, he is losing control of his sword's manifestation.
As we get more zanpakutou lore later on, we learn that shikai or bankai will revert to their sealed form if the user loses consciousness or as they come close to death. In that case, it seems like the mechanism at play is that the shinigami needs a certain amount of reiatsu to maintain their sword's transformed state.
What's going on here is the opposite: there is too much power going into Ichigo's sword. Later on, when Ichigo is training with Urahara, Urahara tells him this about his sword:
Ichigo doesn't know what he's doing. He doesn't know his sword's name. He doesn't know how to put reiatsu into his sword, or how to control his reiatsu at all. I've been referring to it as "his sword", because I'm not really sure it's fair to say that it is actually Zangetsu yet at this point. He's putting too much power into it, and he doesn't know anything about bindings or seals or other ways that you put yourself into a sword and get it to stay there, which are presumably taught and practiced during asauchi training at the Academy. [insert obligatory names-as-bindings rant].
Something that's not really canon but that I like to think about a lot is the idea of shinigami manifesting in the World of the Living. Most people who die in Bleach automatically go to Soul Society (I think some may go directly back into the resurrection cycle, but I'm not positive on that). Some stick around as Pluses/ghosts, and those with strong grudges may turn into Hollows. I think a shinigami has even more of that "sticking around" power than a normal plus, which is how they are able to come back from Soul Society and exist in the Living World. The shihakushou and the zanpakutou (particularly in its sealed form) are physical objects when they are in Soul Society, but when a shinigami is in the Living World, these things are part of their manifestation, sustained by their own reiatsu. I think individual shinigami may be able to do this more or less naturally; it's probably also covered at the Academy. My friend @bleachbleachbleach once joked that the terrible Academy field trip could have been a reward to the Advanced Class for passing their manifestation exam.
Ichigo, on the other hand, is just doing this purely on instinct. He's doing all his shinigami stuff on pure instinct. He makes it look easy, you might say, except for the part where he's doing it wrong, like in one of OP's previous posts, there's a scene where Uryuu notices that Ichigo doesn't actually control his reiatsu.
To get back to the original question of "could any shinigami's zanpakuto go wobbly like this", the answer is "in extenuating circumstances, probably yes" but practically, it's only because Ichigo is both very powerful and has very little training.
It's really interesting that overloading his sword like this causes it to start glitching out. I'm curious as to what would have happened if Uryuu hadn't helped him bleed off his reiatsu. Uryuu assumes he would explode, which is probably not a bad guess. This actually reminds me a bit of the scene in the Don Kanonji arc where the fixed-location ghost disappears and then remanifests elsewhere as a Hollow. I think it quite possible that without Uryuu's intervention, Ichigo might have turned into some proto-version of his Hollow form. If you think about it, this isn't too different from what happens later on in the shattered shaft.
