I understand how a the language of turing machines which do not accept themselves is not recognizable but I'm not sure if the same proof could be used to describe a DFA... i.e a proof by contradiction in which a recognizer M recognizes a TM which does not accept itself on input ...creates a contradiction in which if M accepts then M does not accept itself.
DFAs always terminate, so if you want to check whether a given DFA accepts or rejects a given word (say its representation), just run it on the word. You can even do this in polynomial time.
interesting idea! havent seen this in the literature but maybe its somewhere. what youve asked is a great simpler conceptual stepping stone into more advanced areas of TCS such as undecidability.
in a word: yes! there are multiple ways to do this. for example: choose any enumeration of all the words in the regular language $L$ where $w_i$ is the $i$th word $w_i$ in the language. then change the $i$th single symbol in each word to another symbol in the same alphabet to create $L'$. by diagonalization, $L'$ cannot be regular. this works for all languages that have at least two symbols. also note that in a sense, all regular languages with an arbitrary number of symbols can be "emulated" by an equivalent language of binary strings only.