In the first, "Professor Miller" is modifying "Charles" and is not being acted upon - it's an adjective, and a complement to the direct object, Charles, who is being called Professor Miller.
In the second, "the rabbit" is the direct object - it's being acted upon, that is to say, it's being given. Given to whom? Why to Charles, the indirect object of the sentence.
First, note that "x is y" is not always logically equivalent to "y is x". For example, "Fools are my friends" is different from "My friends are fools" (because the first allows wise men to be my friends too, whereas the second does not); "All men are mortals" is very different from "Mortals are all men" :-)
That said, sometimes there is an equivalence, and some ideas can be expressed either way. In these cases the verb will agree with whichever you choose to make the subject ("One side-effect is headaches"; "Headaches are one side-effect").
(Note that in your examples, "**Our goal were the mountains" is wrong - in standard writing it should always be "Our goal was (complement)". But you could say "The mountains were our goal", with the same basic meaning but different emphasis.)
Addressing your second question: wholesale inversion of a "to be" sentence (where a sentence of the form "noun copula complement" changes to "complement copula noun") is rare, except for in specific situations:
- Questions (because English likes the question word to go first: "Who is Fred? Fred is the tall man"). The inversion always happens unless there's a specific reason not to (perhaps expressing surprise - "He's whose brother?")
- Certain comparative expressions ("Better still are the ones that follow"). The inversion here is optional ("The ones that follow are better still" is fine too).
- Expressions describing location. The simplest in this category are "There is...", "Here are..." and friends, but I'd also include in this category "Next to my house are two restaurants", "Found in every city are cars and buses". This inversion is optional.
- Poetic effect - either for emphasis, or for reasons of metre/rhyme ("Blessed are the meek" - the natural phrasing is "The meek are blessed"; the inversion serves to imply a very great level of blessedness; "Old King Cole was a merry old soul, and a merry old soul was he" - the inversion is clearly unnecessary but nicely fills up the line)
Note that in these situations (except possibly the last), it is usually very clear from the construction that the word or phrase at the start of the sentence is not the subject - often because it's an adjective or adverb phrase. In situations where the distinction is not so clear - such as the examples you provided - inversion will rarely if ever be used (since the sentence tends to end up sounding plain wrong, rather than inverted).
Do note that there are a large number of situations where a slightly different form of inversion ("noun verb complement" changing to "verb noun complement") is used - where it is forced or allowed by the use of certain forms or expressions. Examples such as "Is he tall?", "Never am I angry" are examples of this second kind. I won't attempt to enumerate these, because for one thing there are a lot of them, and for another I don't think the uncertainty you're concerned with arises here. In any case, here is a list of uses of inversion, that contains both types.
[Edit to respond to edit in question]
I have to agree with Fowler about the time periods: the "six months" is considered a single unit, so the singular is used; this is common when referring to measured quantities:
- Ten pounds is a small amount to pay.
- Two litres is more than enough.
We can tell that this is not inversion by using a verb where the ambiguity doesn't arise:
- Six months seems like an eternity.
- Five dollars buys me a very nice lunch.
We can construe the sentence like Fowler as meaning "A period of ...", "An amount of ...", or equivalently by considering the phrase as meaning "Six months of time", "Five pounds of money"; mentioning the uncountable noun makes the reason for the singular clearer, and distinguishes this case from the "light of the stars" case, where there's no obvious way to do the same.
(The plural can sometimes also be used in these cases, giving a sense of referring to each of the individual items mentioned "The six months are dragging on slowly" emphasises that every single one of them is felt.)
Best Answer
Sentence 3: Existential Subjects: Words, Phrases and Functions
[A Comparison of Sentences and analysis of Sentences 1 & 2 forthcoming]
So now the burning question is: What is the subject of this existential sentence? This isn't as easy as it looks. If we want to answer this question we need to really understand what a subject is. And if we want to understand what a subject is, we need to understand the difference between a function in a sentence, and a part of speech, for example a Noun, or type of phrase, for example a Noun Phrase. Being a Noun or Noun Phrase and being a Subject are completely different. A Noun can be a Subject but it can do many other jobs as well. For example, a Noun can also be an Object, a Modifier in another Noun Phrase, or the Complement of a Preposition:
So Nouns seem to be a type of word. The different jobs that a word can do are called functions. The words Subject, Object, Modifier, Complement are all different jobs or functions. There are many other types of function.
Very often we see a group of words, not just one word, doing some particular function in a sentence:
Here, the Subject is two words: The mice. The most important word in that group is the Noun mice. We call the chunk of words the mice a Noun Phrase. We can have very long Noun Phrases:
Here a very keen monkey fanatic has the job of Complement of the Verb BE.
This big Noun Phrase has some words in it that seem to work together. For example the word very seems to go together with the word keen. The answer to How keen is he? is Very keen!. In the chunk very keen, the Adjective keen seems to be the main word. This chunk is an Adjective Phrase. So, we have a big Noun Phrase functioning as Complement of BE, and inside the Complement we have an Adjective Phrase functioning as Modifier. So large phrases have smaller phrases inside them. Every phrase always has its own function.
Sometimes we might want to do a test, to see which chunks of words go together in one phrase. One well-known test is to use a substitute word to replace part of a sentence. Usually, if the substitute word replaces a group of words, then that whole group is one phrase and will have its own function. We can do this test with The mice like cheese. Let's use the word They:
So the word they replaced the words The mice. This shows that the mice is one phrase - a Noun Phrase. It also indicates that The mice has one function - it's the Subject.
There is one very special function we haven't talked about. Us teachers and linguists are a bit naughty; we say things like Subject Verb Object, things like that. Of course Verb is a type of word, not a function! Verbs can have many functions, they can be Subjects, for instance:
But there is one function that can only be carried out by Verb Phrases (in English). No other words or types of phrase can do this job:
Here the job being done by the Verb (Phrase) smoke is the job of Predicate. In the same way that every sentence in English has a Verb, every English sentence has a Predicate. This is the job that the Verb Phrase does. What I want to show you now, is that the functions of Subject and Object are not equal. The Subject is a far more important and fundamental part of any sentence than the Object. Let's do a substitution test with the sentence:
We know that we can substitute they for The mice. Let's use an auxiliary verb as another substitute word. We know that auxiliaries can 'stand in' for other words in sentences. The sentence is present simple so we need the auxiliary DO. Look what happens to the sentence:
We can see that both of the words like and cheese have been replaced. This shows that like cheese is one phrase. The head word is the Verb like, so like cheese is a Verb Phrase. The function or job of the Verb Phrase in the sentence is Predicate.
Inside the Verb Phrase (the Predicate) , there is a small Noun Phrase, cheese. The function of cheese inside this Verb Phrase is Complement of the verb. It is a special kind of complement: an Object. This test shows something very important about Subjects that makes them completely different from Objects and other complements. Subjects are outside the Predicate. We can say that they are external to the Verb Phrase. Other complements of the verb are internal complements. They live inside the Verb Phrase and inside the Predicate. If we replace the Verb Phrase, all the Complements of the verb are replaced too. In fact, everything outside of the Subject is part of the Predicate and will be replaced in a substitution test like this.
Hold on a minute! This sounds like the job of Subject is about how sentences are made, how we build them. What about meaning? Subjects must have a special meaning, mustn't they? Some people say that verbs tell us about actions. It is said that the person or people that do the actions are the Subjects. So in Bob punched Nelson, the Subject is Bob. In this sentence this is true, but in many sentences it isn't. In the sentence Bob is a monkey fanatic, Bob isn't doing anything. The sentence is just telling us about a quality or chracteristic of Bob. There's no action here. Let's do a substitution test for this sentence with the pronoun he and the auxiliary verb BE:
Here we can see that we have a subject and a predicate, even though there's no action and no person doing the action. Ok, so maybe that's not a good test because BE here is used with a stative meaning. How about a sentence like Bob was punched by Nelson. We can ask the question Who was punched by Nelson? The answer will be:
Here we see was being used for was punched by Nelson. Again the sentence has two parts, a Subject Bob and the Predicate was punched by Nelson. Now here Bob wasn't doing the punching, he received the punches. In a passive sentence like this the person doing the action appears in the Verb Phrase. So the semantic role of the Subject in the sentence, what that person does or doesn't do, is not significant in terms of it being the Subject. Notice again that the Subject is external - it's outside the Predicate. We do have names for the person who does something and the person it is done to and so forth. They are called thematic roles. In both sentences Bob was punched by Nelson and Nelson punched Bob, Nelson has the thematic role of agent. Bob has the thematic role of patient.
Subjects, it seems, are indeed about the structure of the sentence. They are about how we build sentences. They are not about meaning, in the sense that they are not about what the Subject is or isn't doing. We have other words to talk about such meaning. Subjects are about grammar, syntax, not meaning.
Let's review what we've looked at. We have talked about how parts of speech and functions are different. We showed that the basic parts of any English sentence are the Subject and the Predicate. Objects and any other complements of the verb are found inside the Predicate. Because the Predicate always has a Verb as it Head, we say that Subjects are external to the Verb Phrase. Objects and any other parts are internal.
Now let's look again at the first example sentence in this post:
We could ask the question Is there one biscuit left? The answer would be:
Here the auxiliary verb is is standing in for the whole Verb Phrase. The Predicate therefore is is one biscuit left. These are the words that have been 'replaced' by is. This shows that one biscuit left is internal to the Verb Phrase. It cannot be the Subject here. It must be a Complement of the verb BE. However, we do still see the word There. This shows that There is not part of the Predicate. It must, therefore, be the Subject! In fact in all sentences like this with expletive there as the first word, there is the subject.
For an introduction to constituency tests and all that jazz, I recommend English Syntax and Argumentation Bas Aarts 2008 3rd Edition.