Well, according to linguistics all sentences have two things: 1) a verb and 2) a subject. If the subject is covert, as is the case with non-finite verbs, it's called PRO (a null or empty pronoun):
In the sentence, "I saw him coming here",
(1) Must a verb (regardless finite or non-finite) ALWAYS have a SUBJECT?
1. I want PRO to visit you.
2. I want him to visit you.
In sentence 1, PRO represents the subject of the verb 'to visit', and in sentence 2, 'him' represents the subject of the verb 'to visit'.
Since 'him' functions as the subject of 'to visit, we expect 'him' to look like a subject, to be in the nominative (subject) form "he", but it is not. The reason for that is related to the non-finite quality of its verb 'to visit'.
Non-finite verbs lack case assigning properties. That is, they cannot assign case to their subjects, so their subjects have to look to the closest verb for case assignment. The verb 'want' is the closest verb, yet it has already assigned nominative (subject) case to 'I', so it can assign only objective (object) case to 'him'. That's why 'him' looks like an object but functions as a subject.
3. I want [him to visit you].
In sentence 3, 'him to visit you' functions as the object of the verb 'want'. Within 'him to visit you', 'him' functions as the subject, 'to visit' as the verb and 'you' as the object.
4. I want [PRO to visit you]
In sentence 4, 'Pro to visit you' functions as the object of the verb 'want'. PRO, the empty or null pronoun, represents the covert semantic subject, 'to visit' is the verb and 'you' is the verb's object.
According to linguistics, all sentence have subjects. Some are overt (we can see/hear them and some are covert (we cannot see/hear them, but we know they are there intuitively. That is, I want and I visit you.) Covert subjects (and objects, too) are represented in linguistics notation by the symbol PRO, which stands for a null (zero) or empty pronoun.
5a. I saw [him coming here]
In the sentence, "I saw him coming here"
(2) Does "coming" have a subject? If yes, which is the subject?
5b. What I saw was him coming here. (object of 'was')
'him coming here' is the object of the verb 'saw'. 'coming' functions as a gerund, a noun that looks like a verb because it ends in -ing. If 'coming' were a verb we'd expect be -ing: is coming, was coming, the continuous form of the verb. There is no verb that sits alone with -ing. That's a gerund. Gerunds, being nominals, cannot assign case, so 'him' gets its case from the closest verb 'saw' but functions as the semantic subject of 'coming'.
6. I saw [him come here].
Sentence 6 follows the same pattern as sentence 5a. 'him come here' is the object of the verb 'saw'. 'him' gets case from 'saw' but functions as the subject of 'come'. If the verb 'come' had case properties, we would expect to see *'he comes here'.
Yes and No. Linguists get around the problem by dividing the sentence into two levels of representation: 1) a syntactic level and 2) a semantic level. 'him' functions as the semantic subject of 'coming' and the entire bit 'him coming here' functions as the syntactic object of 'saw'. In this way, 'him' has a separate function at each level of representation.
(3) Can an OBJECT of a verb be simultaneously the SUBJECT of another verb? "Him" is the object of "saw"; is "him" simultaneously the subject of "coming"?
7a. I saw [him coming here]. (Semantic subject)
7b. I saw [him coming here]. (Syntactic object)
Since 'him' is part of the syntactic object of 'saw', and by itself not the direct object of 'saw', linguists use that positioning to argue that 'him' has a separate function at each level of representation. That is, 'him' is the semantic subject of 'come' and part of the syntactic object of 'saw'.
'John' and 'him' are not the object of the verb 'saw'. They are part of the verb's object:
In the sentence, "I saw John who was coming here" or "I saw him who was coming here",
(4) "John" or "him" is the object of "saw" but not the subject of "coming", because "who" is the subject of "coming". This merely avoids Questions (1), (2) and (3) above.
8a. I saw [this].
8b. I saw [John who was coming here]
8c. I saw [*him who was coming here]
The phrase 'who was coming here' modifies 'John' and 'him'. It's an adjectival clause. The verb of that clause is 'was coming'; 'who', the subject, gets case from 'was coming'; 'was coming' cannot assign case twice, so 'John' and 'him' get case from the verb 'saw'. 'John' and 'him' are the semantic subjects of 'was coming'. 'who' is the semantic and syntactic subject of 'was coming'. 'who' refers to 'John' and 'him' They have an anaphoric relationship: 'who' refers back to 'John' and 'him'.
9a. I saw [John who was coming here]
9b. I saw [*him who was coming here] (Ungrammatical. 'who' refers to nouns, not pronouns)
The answer to your question (4) doesn't 'merely avoid Questions (1), (2), and (3). It's the proof you need to argue that 'him' functions as a semantic subject of one verb and as part of the syntactic object of another verb. The order in which your questions were asked provides a logical flow for argumentation. In other words, I hope you get an "A" :twisted:
All the best,