Yes, the Champernowne binary string is normal (or strongly uniform, in your terms); see also here.

Almost all real numbers are normal, in the sense that the set of the corresponding real numbers in $[0,1]$ is of Lebesgue measure $1$.

Yes, the Champernowne binary string is normal (or strongly uniform, in your terms); see also here.

Almost all infinite binary string are normal, in the sense that the set of the corresponding real numbers in $[0,1]$ is of Lebesgue measure $1$.