Everything is more or less OK, I think.
For ⑩, "entrance" is more natural than "front door" when speaking about a place such as a library. To me, "entrance" is the preferred answer, but I would also consider "front door" to be acceptable.
I think it would also be OK to write something like "near the entrance" or "by the entrance" instead of "at the entrance". A native English speaker might write something like "leading to the entrance", but it is not necessary for your students to learn this.
For ⑨, you wrote "[allowを用いて8語で]". Will students understand that they might have to change the form of the word (i.e., 'allow' -> 'allowed') to get a suitable answer? If there might be confusion, maybe you should write this instruction more clearly.
For ⑦, I think that using "when" is slightly more natural than using "while", but what you have now is fine, so there is no need to change it.