Can a regular language have uncountably many strings?
No, there are only countably many finite strings over any finite alphabet.  Given an alphabet $\Sigma = \{s_0, \dots, s_d\}$, you can associate the string&nbsp;$w$ with the number&nbsp;$1w$ written in base-$d$, giving an injection from the natural numbers to&nbsp;$\Sigma^*$.  (The reason we use the number $1w$ rather than&nbsp;$w$ is to avoid problems with, e.g., $2$, $02$, $002$, etc. all mapping to the same number.)