May I ask, would it be right to say it wants to preserve the old order of traditional Catholic monarchies under the influence of Rome?
In the sense that monarchies were not illegitimate governments, and that it would be wrong to overthrow a monarchy just because one could. The Church accepts many different arrangements of government as legitimate, including the ones as “exotic”, as Republic of Venice with elaborate elections of doge or Polish-Lithuanian Commonwealth with elected king, powerful
Sejm (parliament), legal rebellions (
rokosz) and armed leagues of noblemen (
konfederacja).
After all, it looks like the most relevant part here is “63. It is lawful to refuse obedience to legitimate princes, and even to rebel against them.”.
In effect, a Catholic cannot prefer Robespierre to Louis XVI or Lenin to Nicholas II just because Robespierre and Lenin claimed to rule in the name of the people.
But then, it does not seem to be a reasonable option for any other human with a functioning survival instinct.
One more question, is the separation of church and state thing some contextual thing or is it an explicit condemnation of the very idea?
Usually “Separation of Church and State” ends up meaning that Church is not allowed to have any influence on State, but State can have as much influence on Church, as it wants.
For example, you might note that “55. The Church ought to be separated from the State, and the State from the Church.” cites allocution “Acerbissimam”. This same allocution is also cited for “51. And further, the lay Government has the right of deposing bishops from their pastoral functions, and is not bound to obey the Roman Pontiff in those things which relate to the institution of bishoprics and the appointment of bishops.”. That is, the government used “Separation of Church and State” as a pretext to rule over Church, not as a pretext to stay out of Church affairs.
As you can see, “Separation of Church and State” is not anywhere as symmetrical, as the name would seem to indicate. And yes, it is condemned.
After all, no serious organisation can find something like that acceptable. No party is going to like “Separation of Party and State”, no university is going to like “Separation of University and State”, no trade union is going to like “Separation of Trade Union and State”, no chess club is going to like “Separation of Chess Club and State”.