If a deterministic finite automaton recognizes a language, must that language contain every accepting input? May a DFA recognize a language which is a proper subset of the set of all accepted inputs? 

For instance, if there is a DFA which decides if an input $W$ has $w_0 = w_n$, can it recognize a language $L$ where there is some $\sigma \in \Sigma$ that does not exist in any element of $L$?
No.
The language (only one!) recognized by a DFA is the set of all inputs it accepts.
The definition of a DFA also includes its alphabet, so you have to consider that an intrinsic part of the DFA. If you change the alphabet at all, you now have a new and different DFA, which potentially recognizes a different language.