The ultimate Puritan spiritual classic is The Pilgrim's Progress (1678, 1684) by John Bunyan. It is an allegory for the journey of the individual Christian to God's kingdom, beset by characters like "Mr Worldly Wiseman" and "Lord Hate-good", travelling through places like "Vanity Fair" and "the Slough of Despond". It has had enormous influence in literature and culture. Apparently, it was said that any Puritan household would always possess at least two books: this and the Bible.
Bonus question #1: The idea of 'Praying for the gift of tears' is not original to Puritanism. Evagrius Ponticus, a monk of the fourth century, wrote in his On Prayer (s. 5, trans. Luke Dysinger):
First of all pray that you may receive tears, so that by means of sorrow you may be able to calm the wildness within your soul; and by confessing your iniquity to the Lord, obtain forgiveness from him.
I don't know the answer to bonus question #2.