When did Cuba seize those hospitals? Was it during the initial wave of transition, or significantly later?

The initial wave, but I probably don't need to inform you about the ridiculous degree of development in Cuba's medical system that is still ongoing and still oriented towards free services, public interest, and supporting the health of the third world at least throughout Latin America and Africa.

With China, the argument here is that there seems to be much backsliding in the proletariate's ownership of the means of production, and the reintroduction of landlords.

Edit: Also, I'm not sure how "China actually owns the land under the landlord's building and so the state is the real landlord" is functionally different that taxes, but maybe I'm missing something important.

Like a lot of things about China's laws, this is kind of complicated and you're better off doing your own research rather than idly speculating or asking some asshole on the internet (me), and this is on top of property law already being a complicated subject in general but as an example:

Ownership rights are protected under Article 39 of The Property Law of the People's Republic of China, which gives the owner the right to possess, utilize, dispose of and obtain profits from the real property. However, this right has to comply with laws and social morality. It can harm neither public interests nor the legitimate rights and interests of others.

In general, rural collectives own agricultural land and the state owns urban land. However, Article 70 of The Property Law allows for ownership of exclusive parts within an apartment building, which endorses the individual ownership of apartments.

Ownership of private property is at the mercy of public interest, so should a conflict arise between the two the owners can simply be told to kick rocks (though they are entitled to monetary compensation), which is a significantly stronger version of Eminent Domain compared to, say, America.

Furthermore:

A land user obtains only the land use right, not the land or any resources in or below the land. A land grant contract shall be entered into between the land user and the land administration department of the people's government at municipal or county level. . . .

Article 12 of the Provisional Regulations on Grant and Assignment of Urban State-owned Land Use Right states the different duration of rights provided for different purposes.

The range of durations is from 70 years for a property of personal residence to 40 years for commercial use.