Is “Would rather have had one’s tooth pulled than doing (saying / using)” an idiom or common saying?
Yes, though it would be more often be in the first-person singular, and more often teeth than tooth, since having them several - or perhaps all of them - pulled is obviously a more unpleasant prospect, and hence a greater hyperbole. So, "I'd rather have my teeth pulled".
In connection with this quote, what is the function of ‘as’
It functions as a relative pronoun. It's well attested in respectable writing:
the temper is to be altered and amended, with such things as fortify and strengthen the heart and brain (Robert Burton, The Anatomy of Melancholy)
And wish his mistress were that kind of fruit / As maids call medlars when they laugh alone. (Shakespeare, "Romeo and Juliet")
Know ye not, that so many of us as were baptized into Jesus Christ were baptized into his death? (Romans 6:3, KJV)
A quick check finds one dictionary list it as "now dialectal" so they claim it was once standard English, but now lingers only in some regions. I'm inclined to agree, but note that some other dictionaries have no such warning.
Does it make no sense, if I remove ‘as’ from this sentence?
Worse, it makes different sense. You could though replace it with that or who.
Consider that "I know many of us that..." and "I know many of us..." means something subtly different. The first claims to know many people within the group denoted by us who the rest of the second describes. The second claims to the knowledge that there are many people within the group that the rest of the sentence describes. While that amounts to the much the same thing, it is a different claim to degree of certainty, and the first allows that it could be true of everyone more readily than the second.
Edit: A clearer example might be given by the following true statements:
I know many people speak Cherokee.
This is true. I know that there are 16,400 native speakers of the Cherokee language, along with some others who have learnt it. I do not know any of these people personally, but I know that they exist, and that's what this statement is about.
I know many people speak as speak Danish.
This is true, because I personally know quite a few people who speak that language. This statement is not about the fact that there are people who speak Danish, but that I know some of them.