As Em1 notes in the comments:
She is the subject and is necessary. bore is the verb and part of the predicate and is necessary. It's transitive and requires an direct object: them all. So, all you need is: "She bored them all". Stupid is part of the predicate "to bore (so) stupid" but doesn't add any value.
Using the classifications noted on Wikipedia, "stupid" seems to fit best with "measure":
Measure - Measure adjuncts establish the measure of the action, state, or quality that they modify
- I am completely finished.
- That is mostly true.
- We want to stay in part.
"Stupid" is, more or less, a degree of boredom and isn't a literal state of being stupid. Or, in other words, she didn't cause them to become stupid; she simply bored them greatly.
Let's examine your first example a little more closely:
[1a] Put the cheese back on the table.
You point out that we can omit the prepositional phrase to get a perfectly sensible command to someone pilfering cheese:
[1b] Put the cheese back.
We can't omit the adverb in 1b because
[1c] *Put the cheese
is ungrammatical. The verb to put licenses a mandatory destination.
But notice that we may omit the adverb in 1a if we keep the prepositional phrase:
[1d] Put the cheese on the table.
What's required is thinking in larger syntactic units. What's important is that put requires an adverbial phrase of place, which may be satisfied by back in 1b, by on the table in 1d, and by both in 1a.
The omission of an argument either makes the sentence ungrammatical because the argument is mandatory or it changes the meaning of the head, the syntactic unit governing the argument. (In your examples the head is the verb.)
I didn't tell the full truth about 1c. There is a verb to put that takes a bare object, and in the case of 1c, you may imagine a track and field coach who hasn't been able to find a shot put, but who fortunately has a ball of gouda that his team may practice with. But in that case, the put of 1c means to throw a heavy ball, while the put of the other sentences means to place.
You may see this in your example
[2a] I live in Bristol.
If you omit the prepositional phrase argument, you get the (barely) grammatical sentence
[2b] I live
but that means something along the lines of "I'm alive", which is different from the live of 2, which means to reside.
There's no ambiguity in either 2a or 2b, but the verbs means different things.
Best Answer
Here with murder should be classed as complement. The reason is that the preposition with seems to depend on the verb. Compare this with:
Here the different prepositional phrases we observe can occur freely with a wide range of sentences. There is no sense in which the on in on Monday is there because it is licensed in some special way by the verb CHARGE. The situation is palpably different in the case of charged with. The preposition with intuitively seems to go with the verb.
Now the term complement as described in the Wikipedia excerpt is rather problematic. To illustrate, every direct object is a complement. So for example in:
... the noun phrase headed by smile is regarded as a complement of the verb. However, because the verb SMILE can be both transitive and intransitive (i.e. may or may not take an object), we will still have a workable sentence without the object:
This definition of a complement - as something without which the sentence will be grammatically incomplete - is therefore not very helpful. The best way to understand a complement is as a phrase which fulfils a special kind of grammatical function. A useful way of thinking about this function is that one of the items in the sentence sets up a 'slot' or series of 'slots'. The phrases that fill these slots function as complements of the item in question.
Consider the following utterance:
Here the verb BET sets up slots for the following: a betting adversary (Bob), a stake (£50) and a proposition (Obama will win the election). The items in brackets that fill these slots are therefore complements of the verb bet. We can of course stick an adjunct onto the sentence:
In one of my fits of recklessness, though, is not filling a slot set up by the verb. It is merely tagging extraneous information onto the sentence proper. It is in every sense an adjunct.
When we are not sure if an item is a complement or an adjunct, there are various tests that we can apply. For example, we can replace the verb and its complements with the pro-form verb DO and the pronoun it. If the item is an adjunct then we should still be able to add it onto the end of the sentence. If it is a complement this will (usually) not be possible:
Example (2) shows that in the park is not a complement of the verb, as we are still able to append it to the sentence after the verb and its complements have been replaced. It is an adjunct. Example (3), however, shows that football does seem to be a complement of the verb, because if we repeat it after the verb and complements have been replaced, the sentence is badly formed. We can do the same test with complements of the verb BET:
Sentence (2) above shows that in one of my fits of recklessness is an adjunct. Sentences (3-5), contrastingly, show that the various phrases associated with the verb bet are not adjuncts but complements.
Returning to the sentence in the Original Poster's question, we need to change the pro-forms slightly so that the verb DO is in the passive to match the structure of the original sentence. The following sentences will be useful in terms of comparison:
When the necessary adjustments are made we will see that on Friday and unjustly do indeed behave like adjuncts, but that with murder does not. It behaves like a complement:
All the evidence then seems to suggest that in the example sentence with murder is a complement, not an adjunct.