As you have seen, the syntax of pudet-type verbs is not an easy topic. Here I will limit myself to answering the questions you have included in your post:
- (In the example) "Nōnne tē pudet hoc fecisse? I interpret hoc fecisse as the subject of the verb pudet: is this correct?"
Yes, to the extent that the infinitival clause hoc fecisse (NB: here hoc-Acc. is the direct object of fecisse) can be replaced by a pronoun in the nominative case (e.g. hoc-Nom.: cf. Non te pudet hoc?), your analysis is correct. That is to say, to the extent that your infinitival clause has the same syntactic function as the subject haec (neuter plural nominative) in an attested example like Non te haec pudent? (Ter. Ad. 754), you are entitled to analyze it as the subject.
- Profectō mē pudet hoc ā meō filiō factum esse! In this case, I see hoc ā meō filiō factum esse as the subject of pudet, but maybe I'm wrong. I think these two clauses work as subjects because of the presence of the pronoun hoc which, I believe, is in the nominative case.
Yes, you are right when saying that the infinitival clause can be analyzed as the subject of pudet. However, note that you are wrong when saying that the pronoun hoc is in the nominative case. No, it is in the accusative case, i.e. hoc is the subject of the compound infinitive factum esse. NB: it is often assumed that the accusative case of the subject of an infinitival clause is assigned by the main verb but this is not always the case: e.g. cf. Necesse est te abire or your example Me pudet hoc factum esse. In fact, I would say this situation is more frequent than some people in this site believe...
- So, in the two previous sentences, I see an infinitive clause working as the subject of pudet which conveys the reason for being ashamed and an accusative direct object (tē and mē) that represents the person who is ashamed. Is this interpretation correct?
Yes, your interpretation is correct.
- I'm also trying to understand what is the syntactic structure of the sentences with pudet being used as an impersonal verb, as (I believe) in the first two ones in this post (...) In all these cases, I see a genitive which conveys the reason for being ashamed and an accusative direct object (puerum, quem, me) that represents the person who is ashamed. If I understood correctly, it seems to me that the contents of this other question confirm what I have stated.
Yes, what you have said is in tune with what I said in this previous post.
- So, my question is: are my interpretations about syntax correct? Is
that the usual way to construct sentences with the verb pudet?
As I have just said, your interpretations of the syntax of this verb are correct. The only relevant mistake I have noticed in your post is your saying that hoc is in the nominative case in an example like me pudet hoc ā meō filiō factum esse. As noted, here hoc is in the accusative case since it is the subject of the infinitive factum esse. As for your last question, the usual way (i.e. the impersonal one!) to construct sentences with the verb pudet can indeed be nicely exemplified with the two attested examples you have given from Plautus and Cicero.
To conclude, the usual way to construct sentences with a psychological verb like pudet in Early and Classical Latin is through this impersonal construction (with the Experiencer in the accusative case and with the Stimulus in the genitive case). However, things changed in Late Latin, when the personal construction (i.e. with the Experiencer in the nominative case) turned out to be more prominent. In tune with this change, in Romance languages the Latin impersonal construction with pudet came to be replaced by the pronominal/reflexive one: e.g. cf. Cat. avergonyir-se, Sp. avergonzarse, It. vergognarsi ‘to feel shame’, i.a.
For those of you who are interested in having more information about Latin psychological verbs, take a look at this book. Note that chapter 4 (pages 119-189) is exclusively dedicated to the "me pudet construction":
Fedriani, Chiara (2014). Experiential Constructions in Latin. Leiden. Boston: Brill.