Question

1. He is not as old as me.
(Is 'as' before 'me' a preposition or a conjunction?)

2. He is not as old as I.
3. He is not as old as I am.
(Is 'as' before 'I' or 'I am' a conjunction?)

Answer 1

#1 is grammatically incorrect.

The word "as" here is a subordinating conjunction, and there are a few missing (but understood) words here.
http://grammar.ccc.commnet.edu/grammar/conjunctions.htm#subordinating_conjunctions

#3 is the entire correct sentence. You can see "as" in its conjunction position there, joining the independent with the dependent clause.

#2 is the grammatically correct, shortened form of #3.

Answer 2

This also applies -- http://grammar.ccc.commnet.edu/grammar/conjunctions.htm#correlative_conjunctions -- since there is the "...as...as..." combination going on in those sentences.