1 The swimming pool doesn't necessarily have the equipment for water sports, just allows swimming.
2- I honestly agree with you- peaks are normally found on mountains.
3/4- Hyphen use is often a matter of judgment and different people will use them or omit them in different contexts. They should really try for consistency here, but it could be that one person did the exercises and one the answers.
5- Because the question refers to a specific organ (The organ we wish to tell you about), not organs in general.