Regular set corresponding to regular language
A regular language $L$, is a set over the alphabet $\Sigma$ (that is, $L\subseteq \Sigma^*$) that has a regular expression that generates it. Therefore, by definition - any language of a regular expression is a regular language.
Regarding the language in question, $01$ for example is not generated by $(0+10^*)$, since it is not $0$ and also is not generated by $10^*$ (since it doesn't start with $1$). The same goes for any string starting with $0$, and has more than one letter in it.