In general, the construction 'faire + verb à quelqu'un' means MAKE someone do something. So I would say that your textbook's translation is inaccurate in its notion of 'let', since this would rather be 'laisser + verb' in French. In this light,
1. '...vous pouviez les faire rester au village' simply means 'You could MAKE them stay in the village'.
2. 'faire faire qqch', though this may seem weird at first, would therefore mean 'Make someone DO something'. For example, 'Je fais faire ses devoirs à mon fils'. i.e. I MAKE my son do his homework.
Lastly, as to why one says 'faire lire ce livre A sa fille', whereas we say 'faire rester les jeunes au village', is because in the first, there is an object and a subject. So saying 'faire sa fille lire ce livre' would be incorrect. However, in 2, there isn't actually an object (well, there is, it is actually 'les jeunes', but you can consider this as people). There is no physical object like 'ce livre', but rather an indirect object: AU village. Therefore, you simply say faire rester les jeunes.... This is also why I said 'Fais faire ses devoirs A mon fils', due to the object 'ses devoirs'.
Hope this is clear!