Your assumptions about digit distribution of transcendental numbers are incorrect. There are 2 separate concepts: Transcendental numbers are number like e and π and a=10^-(1!)+10^-(2!)+10^-(3!)+10^-(4!)+... which are not roots of equations with rational coefficients. Normal numbers are numbers (like 0.12345678910111213141516..., although that is hard to prove!) that have every finite sequence of digits (either in their base-10 representation (or some other base) or, if you take the stronger more common definition, in all bases) appearing at the "appropriate" frequency.

It is not known whether π is normal. (See my writeup on Everything2 for details and more links). It is known that many transcendental numbers are not normal. For instance, the number a above cannot be normal, as the only digits appearing in its decimal representation are 0 and 1.