Do there exist numbers normal in every base except for one?

This is not possible. In Kuipers, L. and Niederreiter, H. Uniform Distribution of Sequences (1974), you can find the following exercise, on page $77$:

If $b_1$ and $b_2$ are integers $≥2$, such that one is a rational power of the other, then $a$ is normal to the base $b_1$ if and only if $a$ is normal to the base $b_2$.

(This is also theorem 2.7. in Bailey, D. H. and Crandall, R. E. On the Random Character of Fundamental Constant Expansions, Experiment. Math., Volume 10, Issue 2 (2001), 175-190.)

Therefore, if $x$ is a real number that is normal in every base but one, say $b_1$, then actually it wouldn't be normal to the base $b_1^2 \neq b_1$, contradicting the assumption. The situation is similar for numbers that are non-normal in every base except one.

Theorem 1

If $x$ is normal in infinitely many bases of the powers of $b^k$ then it is normal in base $b$.

Conclusion: This rules out your first curiosity.

Theorem 2:

If $x$ is not normal in base $b^k$ then it cannot be normal in base $b$.

Conclusion: This rules out your second curiosity.

Source for the theorems: On normal numbers, Veronica Becher, page 20-21.

The most general answer to date for your question comes from a paper of Schmidt (http://matwbn.icm.edu.pl/ksiazki/aa/aa7/aa7311.pdf). Schmidt showed that if $A\subset \mathbb{N}_{\ge 2}$ is a set closed under multiplicative dependence*, then there exist uncountably many numbers that are normal to every base $b\in A$ and not normal to every base $b\not \in A$. And by earlier work of Maxfield and Cassels, if $A$ is not closed under multiplicative dependence, then no such number exists. Since if one element is missing from (resp., included in) a set closed under multiplicative dependence, infinitely many are missing (resp, included). Thus, the answer to your question is no.

*A set is closed under multiplicative dependence if $n\in A$ implies every rational power of $n$ that is an integer $>1$ is also in $A$.

I've heard numbers that are normal to some bases but not normal to others be referred to as selectively normal.

As a curiosity, the problem of characterizing possible sets of simple normal bases was recently solved by Becher, Bugeaud, and Slaman (http://arxiv.org/pdf/1311.0332.pdf).