What kind of language is needed to recognize an ordered list? [multihead automata, apparently]
It sounds like what you are looking for are multihead automata (in your case, 1-way 2-head deterministic finite automata should suffice). I'm not really an expert on these, but google turns up some interesting surveys on this language hierarchy, such as

Marek Chrobak: Hierarchies of one-way multihead automata, http://www.sciencedirect.com/science/article/pii/0304397586900939

This also gives one answer to your second question: The hierarchy of n-head automata lies "across" the Chomsky hierarchy.