We know a movie named Despicable Me. I guess it means "I am despicable". But can me be used in such a way?
For example, can I further say unavoidable me to mean "I am unavoidable"? Are there other examples? Or can I just do this with any adjective?
Considering this issue strictly in terms of grammar, there are two things to consider here.
First, in the phrase "despicable me", "me" is a noun (a pronoun, actually, but still a noun) and "despicable" is an adjective. So in one sense, yes, it is perfectly grammatically appropriate to place adjectives before nouns ("red firetruck") and by extension it works to place "despicable" before "me".
Second, the sentence as a whole is a fragment. There is no noun performing a verb; there is simply a lone adjective-noun pair, which technically is not enough to form a full, proper English sentence. However, it could be considered an exclamatory sentence, especially in context, so we let it slide.
In short, it is a proper adjective-noun phrase, and we'll look the other way on the whole fragment thing, so yes, it can be used in such a way.
Recall how people usually exclaim this statement in order to self-sympathize:
The above is another example of the X me statements. Note that they are only used in casual conversations.
Colloquially, yes. Literally, no.
In everyday speech if you wanted to say fashionable me! (or moi for effect) Meaning, I am fashionable, it would be considered idiomatic.
With an exclamation you could even change the meaning:
Not on your life.
However, if you wanted to imply the meaning in writing that you were fashionable, unless you were doing so in direct dialogue (as above), you would be better expressing it as I am fashionable or I am in vogue, I am chic... etc.