In this case "quite" is used as an emphasiser, for example:
Jane is quite tall
Jane is pretty tall
In both of these cases the sentences mean the same thing, i.e. that Jane is tall, with an extra emphasis placed on Jane's tallness.
It's use in this form is uncommon outside of British English, although even there its use is becoming increasingly rare.
Putting this together with your sentence, we can see that "quite the opposite" is merely an emphasis of "the opposite", hence the following two sentences are effectively equivalent:
"short" is the opposite of "tall".
"short" is quite the opposite of "tall".
Consequently in your sentences, sentence 1 and 3 are both perfectly fine either with, or without the word "quite".
Sentence 2 and 4 are not quite right. "Opposite" tends to be used comparatively, for example:
"left" is the opposite of "right"
Whereas "contrary" reverses the meaning of a sentence and restates it:
Is Jane still short?
No! Quite the contrary! (Now she is tall.)
Notice here that "contrary" can be combined with "quite" to give the sentence additional weight and emphasis. "Quite the contrary" is an idiomatic expression that means "No! Exactly the opposite!"
Therefore in answer to your question, "quite" is an emphasiser, and sentence 1 and 3 are correct. Sentence 2 and 4 are not correct because "contrary" cannot be used comparatively.
"Needs be made", in the context of your first quote, is just wrong, and most likely a typo in the comment. You're right, it should be "needs to be made".
However, "need be made" in the other three quotes is correct, albeit more formal and fairly rarely heard. Depending on the context, it can be equivalent to "needs to be made", "needed to be made" or "need to be made". In fact, it can probably cover any tense of "to need", and is something that the vast majority of people will be fine with just understanding it when it's used and not using it themselves.
Hope it helps :)
Best Answer
Both versions are perfectly grammatical and natural-sounding, but they mean slightly different things. Sort of. If you want them to.
This means that some attempt needs to be made towards cleaning the sofa. Unless it's your mother saying this to you, the implied end result is not necessarily a perfectly clean sofa. (If it is your mother, then you're usually better off assuming she wants perfect cleanliness, even if all she says is "Hmm, that sofa has been cleaner." But I digress.)
Here, there is at least a slight implication that the desired end result is a clean sofa, not just a cleaner sofa.
Now, keep in mind that these are just-barely-discernible shades of meaning, not outright "use this if and only if you mean..." In practice, the intended meaning is almost always total cleanliness or a reasonable facsimile thereof.
Regarding 2a and 2b, both are possible1, but neither is what I'd say in that situation. Better alternatives:
1 The version without the "to be" starts to sound a bit like the Central Pennsylvania dialect "that car needs washed" type of construction; it sounds awkward, but not necessarily wrong. But then, I live near Philadelphia, so perhaps I'm not the best judge.