Why was this chapter drawn so crudely? Was it intentional, as a stylistic choice, or had he injured his hand at the time?
On page 14, who's the guy who says "you have a guest" who looks sort of like how Kite/Kaito did before his reincarnation?
This chapter is somewhat infamous for its bad art, actually. Usually Togashi cleans up these kinds of chapters in the volume releases, but he didn't do so for this chapter in particular.
The Chimera Ant arc was a very painful time for Togashi. It took him 8 years to complete and was fraught with constant hiatus (unfortunately all too familiar at this point). Considering how close this chapter was to the end of the arc, he was probably exhausted and in constant pain. So yes, he was suffering from injuries, hand or otherwise.
However, I believe leaving the art in this rough state was also a stylistic choice. I think this quote from page 19 expresses his feelings as a bit of meta-commentary: "Wanting things to be easy and trying to run from everything. Even if it's painful, I'll try to live. So when I die, I can say 'I gave it my all'."
disclaimer: a lot of this was inspired by some YouTube video I watched but I can't remember who made it.