Now, it is true that Cardinals could be (and have been) laymen ... and don't have to be bishops or priests or clerics.  But women?  And I don't think that lay Cardinals can vote in conclaves.

That's not true.  In order to hold any ecclesiastical office, you have to be a cleric.  That means you have received tonsure and have entered into the clerical state and thus have become a member of the Church's hierarchy.

The events of interest in Card. Lienart's life:

Born, 1884
Ordained, 1907
Became mason, 1912 (supposedly)
Promoted to the 30th degree, 1924 (supposedly)
Became Bishop, 1928
Ordained Archbishop Lefebvre, 1929
Became Cardinal, 1930

An interesting analysis by Dr. Coomaraswamy.

If a laymen were appointed Cardinal, he would indeed be tonsured, but not even the minor orders were required.  But clerics are not per se members of the "hierarchy".  1917 Code of Canon Law required that Cardinals must be at least priests.  But even being a cleric is not required by divine law, since the Candinalate is not of divine institution.

"Hierarchy" is not a univocal term.  It can mean jurisdictional hierarchy or ecclesiastical hierarchy depending on context.  In the absence of context it should be assumed that the more general term, ecclesiastical hierarchy is the intended meaning.  The ecclesiastical hierarchy consists of all Catholic clerics, regardless of orders.  The jurisdictional hierarchy is more specific and consists of all those clerics possessing an ecclesiastical office with ordinary jurisdiction attached to it.  I don't know how relevant to the discussion this information is but I brought it up because I think a misunderstanding of the definition of hierarchy is a contributing cause of the current crisis.  SVs are getting hit with the accusation that the theory would mean the destruction of the hierarchy.  The jurisdictional hierarchy is indeed greatly diminished if not completely destroyed.  But the ecclesiastical hierarchy remains.  And the ecclesiastical hierarchy can restore the jurisdictional hierarchy if they put their minds to it.

In either case, is it somehow incompatible with Divine Law to have a lay Cardinal?  I don't necessarily see that.

If by "lay Cardinal" you mean a man appointed to a Cardinalate who is not yet ordained, yes, it has happened.  It even happened for the papacy.  But the man cannot receive the office until he has received tonsure.  After which he can immediately exercise the jurisdiction attached to the office (even before he is ordained).  But to call that a "lay Cardinal" is an abuse of terms.  He is just a Cardinal and all Cardinals are clerics.  You can make a distinction between a Cardinal who has no orders and one who does have orders.  But neither one is a layman.

But is this of divine institution?

I think it would be more proper to say that it is by definition of the hierarchy that all of the members are clerics.  And the hierarchy is certainly a divine institution.  http://www.newadvent.org/cathen/07322c.htm

 Ordained priest in the 1980s by Bishop Lefebvre. Consecrated bishop on December 27, 1992 by Bishop López-Gastón. In 1997 he suspended his duties as priest and bishop due to his doubts about the validity of his priestly ordination ...

So, what woman is he living with now?

Yes, but Lienart ordained +Lefebvre to the priesthood on his own, and if +Lefebvre wasn't a valid priest, it's doubtful that he could be validly consecrated a bishop.

Nevertheless, there's no doubt whatsoever about his validity.

Bishops were created first by Our Lord at last Supper. All twelve Apostles were laymen and were ordained Bishops straight away. It is only later on, as the number of converts increased, that the Apostles instituted "lower" Orders of the Sacrament, which would receive delegated powers to perform some of the duties of the Bishops. These were the Diaconate and the Priesthood. So, in theory, a bishop could consecrate a layman bishop straight away, and it would be valid. In other words, there is only ONE sacrament of Holy Orders instituted by Our Lord, and that is the Episcopacy. The lower orders came later by the institution of the Church.