The original question mentions that given a commutative ring R and an ideal I, two topological spaces (defined using the Zariski topology) are homeomorphic. My answer was addressing the question ``what is an analogous statement for the etale topology?'' (The answer being that you can extract two topoi which are equivalent.) I don't think you can formally deduce anything about defining sites from this. Rather the information would go in the other direction: to write down a proof of my claim, you'll want to think about the problem of lifting etale coverings, as in your earlier reply.

The homotopy category definitely doesn't embed fully faithfully. Maybe the easiest example (working over a field C) is to take the usual polynomial ring R = C[x,y]. In commutative dgas this is already cofibrant, so [R,A] = H_0(A) x H_0(A) for any cdga A. In associative dgas a cofibrant replacement for R is the free associative algebra on x,y and z in hom. degree 1 with dz=xy-yx. So if A happens to be a cdga you get [R,A] = H_0(A) x H_0(A) x H_1(A).