Is the existence of qualia generally considered an unresolved problem by philosophers? Is there a consensus on its nature or whether it can be studied at all?
Purely philosophically, consciousness is stuck between two schools of thought (generally, since each idea also have different sub-theories).
See more on the wikipedia article on consciousness, especially the mind-body problem.
One is the materialist/realist idea, where consciousness is a product of cause and effect by random events dating back to the beginning of the universe with the big bang.
The other is the theist/deist/idealist idea, where consciousness is something beyond the physical, and has the power to influence our brain. Each school of thought presupposes something beyond the idea itself. Materialism presupposes that there is just matter and energy across time, and nothing beyond that. The converse presupposes that there exists some metaphysical higher plane outside of the purely physical, and that our mind/consciousness is either completely or partly residing within this higher plane, interacting with the physical plane.
The train of experiments that had it's outspringing in the original double-slit experiment seems to suggest that the mind is not only responsible for influencing the brain, but that it is quite literally responsible for manifesting it and the world around us. The mind might actually be more real than matter. In a way. InspiringPhilosophy on youtube has a video about this, where he goes through and explains the studies, with clips from interviews with experts.
I am not sure if the issue is resolved or not though. I think we are getting close.
Consciousness is not a problem in philosophy. It is only a problem in university philosophy, where it is not studied. The study of consciousness is called Mysticism or Yoga. Academics just talk about it and argue about their speculative theories.
Nobody in the perennial tradition talks about the 'problem of consciousness'. Rather, they talk about the problem of ignorance.
An answer to your question would require giving a bit more detail since the problem has many aspects and may be formulated in various ways. Chalmers' 'hard' problem, for instance, is easy to solve, but other related problems are more tricky.
See https://qualiaresearchinstitute.org. It has very interesting things about this topic. But from just researching, it is trying to be resolved, but is still unresolved. Otherwise, I think science will help more with question than just pure philosophy.