To take the last question first: no, the forms in the examples in the question here are not gerunds. They are verbal nouns.
Morphologically speaking, gerunds and verbal nouns are indistinguishable in English: they both end in -ing and are identical both to each other and to the present participle (which is a different kettle of fish that I’ll leave out of this).
The only way to tell them apart is by their syntactic properties (but not, as rogermue surmises in his answer, their syntactic function): whereas a verbal noun is a true noun and functions as a true noun, a gerund has both noun-like properties and verb-like properties:
- Verbal nouns can: be modified by any kind of determinative (and possessive determiners1 included) and adjectives, be pluralised, modify other nouns in of-constructions. They cannot: be modified by any kind of pronoun or by adverbs, take subjects or objects
- Gerunds can: take subjects (NPs, objective pronouns, or possessive determiners) and objects, be modified by adverbs. They cannot: be modified by other determinatives (like articles), be pluralised, modify other nouns in of-constructions
The two types are in complementary distribution, then.
The grammatical examples in this question are all instances of verbal nouns, rather than gerunds: they are preceded by determinatives (and we can switch the possessive ones for definite articles without losing grammaticality) and adjectives; and they modify their ‘object’ by using of-constructions.
The ungrammatical examples, on the other hand, attempt to shoehorn a verbal noun into being a gerund by having a pronoun precede it, though maintaining the modifying adjectives and the of-constructions. Ungrammaticality naturally ensues.
If we convert the ungrammatical examples into actual gerund clauses by also substituting adverbs for adjectives and direct objects for of-constructions, they become completely grammatical:
√ His continuous meddling was starting to bother me. →
× Him continuous meddling was starting to bother me. →
√ Him continuously meddling was starting to bother me.
√ The senate characterized their sinking of the flagship as rash and excessive. →
× The senate characterized them sinking of the flagship as rash and excessive. →
√ The senate characterized them sinking
of the flagship as rash and excessive.
Note that, while both options (verbal noun or gerund) are equally grammatical, one is frequently preferable to the other, even in syntactically similar circumstances: the choice depends on a variety of factors, but most basically I would say it is a semantic one. The gerund expresses the simple fact that the verbal action was carried out, whereas the verbal noun tends to express the manner in which it was carried out. So if the manner in which something happens that is important, the verbal noun is preferred; if it’s the simple fact of the action taking place at all that’s relevant, the gerund is preferred; and if either or both is relevant, both work equally well:
√ Him continuously meddling [the fact that he meddles continuously] was starting to bother me
√ His continuous meddling [the continuous manner of his meddling] was starting to bother me
√ Him preparing the roast dinner [the fact that he was the one preparing it] was disappointing
√ His preparing of the roast dinner [the way he prepared it] was disappointing
√ His skilful engraving of the vase [the way he engraved it] was unparallelled.
(×) Him skilfully engraving the vase [the fact that he engraved it] was unparallelled.
√ Him downing that two-litre pitcher of beer [the fact that he was able to do it] in 4.8 seconds was the best thing I’ve ever seen
(×) His downing of that two-litre pitcher of beer [the manner in which he did so] in 4.8 seconds was the best thing I’ve ever seen
As the examples above show, the answer to the first question is never: a gerund that expresses a subject can always do so through an objective pronoun. It is only when dealing with verbal nouns that objective pronouns are not an option.
On the other hand, it is frequently the case that the subject of a gerund can be expressed by an objective pronoun, but not idiomatically by a possessive determinative. In other words, the exact opposite of what the question here asks. In this case, the difference is, as far as I can tell (I’m sure GaGEL deal with this in more detail, but alas I have no access to it), based on what components the gerund clause contains:
- Gerunds which are not modified by adverbs can freely licence either possessive determinatives or objective pronouns to represent their semantic subjects
- Gerunds which are modified by adverbs overwhelmingly prefer objective pronouns to represent their subjects
In other words:
√ I appreciate you taking the time to help me.
√ I appreciate your taking the time to help me.
√ I appreciate you voluntarily taking the time to help me.
(×) I appreciate your voluntarily taking the time to help me.
My gut instinct is to call the last example entirely ungrammatical, but I have a hunch some would consider it acceptable.
This actually implies that there are really three levels in the verb↔︎noun hierarchy of non-finite verbish forms:
- Verbal nouns are pure nouns. They construct NPs and as such, their subjects must fill the determiner slot in the NP and thus be determinatives.
- Unmodified gerunds are on the fence: they can either be verb-like and form non-finite gerund clauses, allowing direct objects and licencing full NPs (including pronouns) as subjects; or they can be noun-like and form NPs, taking determiner subjects, though they retain the verb-like property of allowing direct objects.
- Gerunds modified by adverbs are the most verb-like. They form non-finite gerund clauses, freely take direct objects, and require NP subjects.
(1) thus has no verb-like properties at all, only noun-like properties; (2) has the verb-like property of taking direct objects and optionally also NP subjects; and (3) have virtually no noun-like properties, allowing direct objects and requiring NP subjects, like finite verbs.
1 Note that ‘possessive determiners’ here refers to any possessive expression that can fill the determiner slot in a NP. That includes the possessive determinatives (my/your/his/her/our/their), but also NPs with possessive clitics (Bill’s, the King of Spain’s, etc.).