Learn English – Should the preposition “of” always follow the verb “to approve”

Q1: Is it jealousy of sth or jealousy at sth or jealousy about sth when referring to the reason why someone is jealous?

I would think you would use the same prepositions for jealousy as for jealous.

Paul was jealous of Bill. Paul expressed jealousy of Bill.

Paul was jealous for/over his wife. Paul expressed jealousy for/over his wife.

Paul was jealous for/over Bill's car. Paul expressed jealous for/over Bill's car.

_{Somebody correct me if I'm wrong, but I believe you can use either for or over with jealous in either of the latter two senses it holds above.}

I have never heard that you could be jealous at something or about something, but if the usages are valid, I would expect that you could use them in the same with jealousy.

Q2: Is it common to say jealousy of sb when referring to the person of whom I'm jealous?

Jealousy of/for/over/at/about something is awkward and in my experience little-used. It is more common to say, "I am jealous of him," as in the example first sentences above.

"A transitive particle verb" is also called a "transitive phrasal verb" or a "transitive two-word verb". It's the kind talked about in the Q about "looked at". Wikipedia says: "A transitive particle verb has a nominal object in addition to the particle. If the object is an ordinary noun phrase, it can usually appear on either side of the particle, although very long noun phrases tend to come after the particle". So, when it's possible to say something like Switch the light off, "the light" appears to function as the direct object of "switch". And when you say "say something like Switch off the light," "the light" appears to function as the direct object of "switch off".

I use appears to function because I don't want to make an authoritative claim here. This requires the expertise of a professional linguist because it's a small technical point, not something that seems to have any effect on how we civilians (including "lapsed linguists" [like me], as one user here calls himself) use the language. All we need to know is whether we can separate the verb form switch and the particle/preposition (whichever your theoretical position says the term should be) by putting the light in between them.

I'd say the answer to your question is "No". But there are other verbs of this type, e.g., hand in as in {Hand in your homework / Hand your homework in} before the test.