Are Cellular Automata always computers?
"... is it always a given Cellular Automaton is a computer?"...


I interpret "is a computer" as "being capable of universal computation" or in other words "being capable of simulating an arbitrary Turing machine".

The answer is clearly NO; a trivial example is a CA that doesn't modify the input:

0[0]0 -&gt; 0
0[0]1 -&gt; 0
0[1]0 -&gt; 1
0[1]1 -&gt; 1
1[0]0 -&gt; 0
1[0]1 -&gt; 0
1[1]0 -&gt; 1
1[1]1 -&gt; 1


(rule 204)

According to Wolfram's (qualitative) classification, only type-4 CAs exhibit a behaviour that can potentially be "universal". But it is an informal statement (like many others in Wolfram's NKS) ... in order to say that a particular CA is/is not Universal you must prove it formally.

For a more rigorous introduction to the argument I suggest you to read some research articles on the subject, for example: Kristian Lindgren, Mats G. Nordhal. Universal Computation in Simple One-Dimensional Cellular Automata. Complex Systems 4 (1990) 299-318

Or - for a good, more general, introduction to computational universality - read Chapter 7: The Grand Unified Theory of Computation of the (IMO excellent) book:  Cristopher Moore, Stephan Mertens. The Nature Of Computation (2011)