Probability that a leaf with 1 will be selected in game tree evaluation
When you talk about probabilities, you should first decide what is actually random. Note that there are two possible sources of randomness here, the value of the leaves (i.e. were talking about some distribution over the possible inputs), and the random choice of what leaf to evaluate first (i.e. inner randomness of your algorithm).

When we say our algorithm is randomized, we usually mean the latter. The complexity of such an algorithm is considered with respect to the worst possible input. More formally, saying that the algorithm runs in expected time $T$, means that $T=\max\limits_{i\in I}\mathbb{E}\left[T(i)\right]$, where $I$ is the set of possible inputs and $T(i)$ is the running time on input $i$ (which is a random variable, since we have a coin-tossing algorithm). This is a worst case complexity measure, as the probabilistic analysis is done relative to the worst possible input. Here the worst case (conditioning on the output being $1$) is that exactly one leaf evaluates to $0$, and in this case the expected number of evaluations is indeed $\frac{3}{2}$.

The analysis you described corresponds to the former option, i.e. we are interested in average-case complexity, relative to the uniform distribution over possible inputs (so we actually have two sources of randomness here). In this case, conditioning on the output being $1$, you are right to say that the value of a leaf is $1$ with probability $\frac{2}{3}$, which makes the expected running time $\frac{4}{3}$ (where the expectation is taken over the leaves values and the coin outcome).