Expectations in Gali's classical monetary model
As for your first question, we are not just assuming $\partial U/\partial B &gt; 0$. We simply assume monotonicity of preferences, which for the model here is a plausible condition. In simple terms this means that "more is better". Hence, the consumer will always choose to spend all of his budget for current or future consumption, given by $C$ and $B$ respectively. Hence, the constraint holds with equality. In these types of models we pretty much always assume equality of the budget constraint.

As for your second question, solving without expectations and then taking the expectation is strictly speaking incorrect. It will give you the correct result here though. What you can do here is to take the derivative in the expectation. For these models that usually works in the non-stochastic setting as here. If you have for example stochastic income with log-normal distributions you would have to be more careful. What you can do then, depending on the case, is solve the expectation before taking the derivative or use the Leibniz Rule.

Hence for your problem you don't forget the expectation in the first place, which corresponds to $\partial U / \partial C_{t+1}$. Instead for your model you can take $\partial E_0[U] / \partial C_{t+1} = E_0[\partial U / \partial C_{t+1}]$. As you can see, it will yield the same result as simply taking the expectation later on $\partial U / \partial C_{t+1}$. However, that is the wrong way and you'll probably lose points in an exam if you write it down that way.