What is the range of values of the expected percentile ranking?
What is $t$?


It means observed $r_{ui}$ in the one-week test set (page 6-left).


  Is my understanding of the metric correct?


First two examples are correct. Assuming user-item relation $r_{ui}^t$ is constant $a$ for all items in the test set, and predicted ranks are uniform across $[0, 1]$, then, the third one would be:

$$\overline{\text{rank}} = \frac{\sum_{u,i} r^t_{ui} \text{rank}_{ui}}{\sum_{u,i} r^t_{ui}}=\frac{\sum_{u,i} a \text{ rank}_{ui}}{\sum_{u,i} a}=\frac{1}{|I|}\sum_{u,i} \text{ rank}_{ui}=\frac{1}{|I|}\frac{|I|}{2}=\frac{1}{2}$$
This makes sense. Items are identical to the user, therefore no model can do better than random guessing, since there is no observed preference to help the model favor one item over the other. Of course, another assumption here is that training (4 weeks) and test (next week) sets are from the same distribution.