Can you recommend the hotel in which I can stay?
Grammatically correct. As I mentioned in a comment before I started writing this answer, you would be unlikely to say "THE hotel" as "the" indicates there is only one. You would be more likely to ask, "Can you recommend A hotel ..."
This is the box in which the book is kept is very old.
Incorrect. There are two prepositions, "is kept" and "is very old". Depending on what you are trying to say is old, you might mean, "This is the box in which the book that is very old is kept" or "This is the very old box in which the book is kept". Or "The box in which the book is kept is very old". Etc, one could come up with other variations.
This is the best place in which I have visited.
Incorrect. We don't say "in which I have visited". The "in which" serves no purpose as there is only one verb in the sentence, "have visited". You could say, "This is the best place I have visited".
I will show the house in which I lived when I was young.
Correct. I might note that a native speaker would be unlikely to use the wordy "in which". He'd be more likely to say, "I will show the house I lived in when I was young."