I've been thinking about the following and I think the answer's in the affirmative.
Is it true that every subset of a DFA-acceptable language that is regular is also DFA-acceptable?
No. Counterexample: Alphabet is numbers digits. DFA accepts all natural numbers. Subset: DFA accepts all prime numbers.
Edit: Alphabet is digits. Sorry, wrong terminology there.
Natural numbers can be expressed as a regular language (and therefore a DFA can be constructed for them):
0|([1-9][0-9]*)