how to view conditioning on a random variable rather than a particular value of that variable The question is about $P(Y|X)$ versus $P(Y|X=x)$. Is there an alternate way to write $P(Y|X)$
that makes its meaning more clear?
I believe these are correct equations for conditional entropy:
$$
  H(Y|X=x) = -\sum_y P(y|x) \log P(y|x)
$$
$$
  H(Y|X) = -\sum_x P(x) H(Y|X=x) = E_X \left[ H(Y|X=x) \right] = -\sum\sum P(y,x) \log P(y|x)
$$
If these are correct, it shows writing $H(Y|X)$ instead $H(Y|X=x)$ implies an additional expectation over $X$.
It makes me wonder if $P(Y|X)$ can be written as $E_x P(Y|X=x)$. I guess that is wrong, but do not know.

It's more a notation mess...
$H(Y|X)$ is the entropy of the whole distribution, varying the second term, where instead $H(Y|X=x)$ is just the entropy of a specific distribution... for example, suppose that $X,Y$ are Gaussian and they are correlated ($N(\mu, \Sigma)$); then:

*

*$H(Y|X)$ is the entropy of all the possible Gaussian over $Y$ that you can have by fixing/observing $X$ (in other words, take an X, consider the Gaussian that forms for Y, calculate the entropy, then move to another X etc etc)

*$H(Y|X=x)$ means that instead of considering all possible observation, you consider only the Gaussian that you have on Y when you observe $X=x$
For the probabilities instead, they mean the exact same thing, $P(Y|X=x)$ is just making explicit what value is $X$ taking
