But, in some grammar books, the second "as" is thought to be a conjunction. Is it? Thanks!
Yes, traditional grammar analyses the second "as" as a conjunction because it introduces subordinate clauses.
But modern grammar rejects that analysis and calls it a preposition. After all, it is no different syntactically to the "as" in my example (4).
In other words, it's just a matter of complementation: it makes no sense to call "as" a conjunction when it introduces clauses like "as I do", but a preposition when it introduces noun phrases like "him".
I wouldn't worry too much about the different word categories. The most important thing is to understand the structure of comparative clauses, in particular the obligatory 'reduction' that I mentioned.