FWIW the equivalence between NFA and DFA requires an exponential space increase to encode the NFA as a DFA, with an exponential space blow up you can encode a lot of things as DFAs (I'm pretty sure you could encode a Turing machine that uses bounded space on the tape as a DFA with exponentially more space, "just" make each possible configuration one state in the DFA)
All three of these representations are capable of describing any regular language (set of symbol sequences, or more intuitively a set of strings), and the fact that a language can be described by an NFA, DFA, or RE implies that it is regular.
I am not hugely familiar with Pearl's "extended regular expression" system, however I was under the understanding that the set of languages it can recognize is a superset of the set of all regular languages. Based on [2], it would appear that Perl regexes can recognize all regular languages, and parts of the set of all Turing-recognizable languages.
0 - Introduction to the Theory of Computation 3/e, Michael Sipser, Thm 1.39, pp. 55.
1 - Introduction to the Theory of Computation 3/e, Michael Sipser, Thm 1.54, pp. 67.
2- https://www.perlmonks.org/?node_id=809842