Christians sing hymns from the book of psalms, or hymns written by other song writers, and they sometimes recite the bible(but not singing, like muslims do). To the second question, no I don't think there is such a thing. At least not officially. 
Hmmm I see, I always thought they were like us repeating the words of God or at least the words of Jesus. By the way, regarding our recitation, in fact this is called in Arabic "Tarteel/Tajweed" it's a special way to read the Arabic Letters with it's basic rules. As you know Arabs are popular for poetry and they have special way to read the Arabic Poetry, Nasheed (It's almost like a Hymn without any instruments) and finally the Quran. When you read the Arabic the pronouncing and the ending of each verse will be similar unlike when you read the translation, hmmm don't know how to explain that but some Arabic words to pronounce it we must increase the length of the character. For example the characters "Alif Lam Mem" there are some rules in Arabic that Alif can only be pronounced Alif while the Mem should be pronounced Meeeeeeeeeem because of the position of the character. For example check this recitation at the second 50 >>>
He is reading the word "Samawat" meaning heavens/skies but you see he is increasing the length because the word in that position requires you to length but it's not a must that's why some other make it 2 seconds while others 1 while others 3 ^_^
Here is another >>>
As you have realized at the end of each breath he is increasing the length
LoooL I know that's complicated! That's why sometimes you will find a man reciting Quran normally like if he is reading a newspaper while some others read it like the links I gave you.
Hmm as for the rewards, maybe that's why we feel that we Muslims are lucky, I mean our prophet Mohammed said that even smiling is considered a "Good deed" that you will be rewarded for ^_~
Jews tend to cry at walls...close enough
I saw one of them were holding a book I guess the Old Testament and was reading it almost like the singing. I don't know if that was their prayer or he was only reading the book?