Is a particular string regular (e.g is '010') regular?
Being regular is a property of languages, not of words.

However, it seems that by '010' you really mean the language consisting only of the single word '010', that is, $\{ 010 \}$. This language, just like any other finite language, is regular.

Where does your pumping lemma proof fail, then? Here is a complete statement of the pumping lemma.


  If $L$ is a regular language then there exists a constant $n$ such that for all words $w \in L$ of length at least $n$ there is a decomposition $w = xyz$, where $|xy| \leq n$ and $y \neq \epsilon$, such that $xy^iz \in L$ for all $i \geq 0$.


Your argument is ignoring the constant $n$, which could be larger than the length of all words in $L$.