1) Are the phenomena that need explanation the same in the hard problem as in the soft problem?
There are some cross-overs.
2a) If the answer to 1) is no, what phenomena need explanations in the hard problem that don't in the soft problem, and vice versa?
The ultimate answer to the hard problem would explain things like:
Why we aren't philosophical zombies.
The origins of consciousness.
Whether the experience you and I both label red, is in fact the same experience, not just the same label.
Is a system of humans, which functions identically to a human brain, conscious?
What is the function of qualia?
Solutions to the soft problem merely measure correlation. So if the patient reports the sensation of red, that whatever set of physical circumstances that correlates to that report, cause the sensation of red. The sensation of red is never measured.
3) Can you conceive of anything that is neither physical nor mental?
4) Can you conceive of anything that is both physical and mental?
Take something mental, and something physical, add them to a system. That system is now both mental and physical.
For both 3 and 4, I obviously cannot conceive of anything not-mental, as conception is a mental process. I allow for the possibility outside of my mind though, which is why I gave those answers.
A successful parallel is life outside of Earth. We need to prove there is not mental activity outside of neural networks as much as that there is not life outside of Earth. Whether we will be able to, is a different question.
Again, how can two people share an experience of a physical entity, when the experience was a mental entity? One cannot experience a physical entity, instead one has to divert to representational theories of perception, which have similar flaws.
To disprove Idealism, one needs to provide a situation that cannot be accounted for with purely mental phenomena.
We agree on Solipsism.
Secondly, Idealism does not eliminate the soft problem. However, if existence is equated to consciousness, then why consciousness exists is answered by a tautology. Eliminating the hard problem.
Granted, Idealism as in the first paragraph, does not provide the second paragraph's solution.
Solipsism is a similar case.