Does mixed-mode operation qualify as a QSO?
I'd say absolutely, yes. You're making contact with another amateur radio station on a frequency allocated to amateur radio; in my book, that qualifies as an amateur radio contact or QSO.
What is the legal status of such a cross-mode QSO?
Assuming that your license allows you to transmit on the frequency and mode that you are transmitting on, I see no reason why there would be any legal problems with such a QSO. The law can only apply to what you transmit, not to what you receive.
It might be possible that in some jurisdictions there are restrictions imposed on what kind of receivers individuals may possess, but even so, I think we have to assume that your possession of the receiver is legal; if it is not, then that's a completely separate problem.
How would it be entered in the log?
I would enter the details of my transmission, with a suitable comment in the comments field for the received signal. For example, station A would write mode CW, comment "received SSB" or something like that; station B would write mode SSB, comment "received CW".
One large part of the rationale for this is that if anyone ever complains about your station operating (for example, due to interference), what will matter is your transmissions, not what you received.
Again, certain jurisdictions may have specific requirements. If that is the case in your situation, you should refer to your local regulations for guidance on what exactly to write into the station log. That said, I would expect any such regulations to be centered around the own station's activities.
Is this situation covered by a regulation?
I doubt it is, although you'd have to consult your locale-specific regulations to be certain.
It may complicate things slightly if you want to use that contact to apply for a mode-specific award, but that's about the only complication I can think of. If you want to claim such a QSO for credit toward a mode-specific award, you'd have to contact the issuer of that award and ask how exactly they will count it.
And of course, the QSL card cannot say for example mode 2xCW or mode two-way CW because that would be inaccurate. For a cross-mode contact, I'd probably make it explicit which mode was used by which station, for example by writing mode CW(CL1SGN), SSB(CL2SGN).