If you understand the graph traversal algorithms in question, there's nothing to memorize. You can just think through the problems. Is it important to understand graph traversal algos? I don't think it is for everyone. But a lot of people who like CS find it interesting. And if you work on, say, a web crawler or a game AI, then you'll need them.
The last two questions are little more than "what is the name of the algorithm I should use to solve this problem?".
This doesn't test understanding in any way. How is it useful to determine whether someone can work on the examples you give, let alone the average software engineering job?
The goal of this quiz is to be a little challenging and fun and maybe educational. It's not to judge whether someone can do a software engineering job!
The answers to the graph problems are indeed fairly straight forward. But asking about why one might use a BST in place of a sorted list prompts interesting discussion. And going into Dijkstra's vs A* and iterative deepening depth first search is (I think) pretty interesting!
