All of them except numbers 1 and 3 were guessed by someone or other.
Here are the answers. The links for numbers 2 and 4-10 might play music at you: turn your speakers down before you click.
1) Till at last the sun rose blood-red in the morning sky
"Thorns in the Straw" by Graham Kendrick. It was my favourite poem in the_maenad's survey.
2) Sorrow and love flow mingled down
"When I survey the wondrous cross" by Isaac Watts. The tune I usually use is called "Rockingham (Miller)".
3) I give you my will in joyful obedience
"Lord of creation, to you be all praise" by Jack Winslow. I cheated here, because the bit I quoted is the latter half of one line and the former half of the next one: "Lord of all power, I give you my will / in joyful obedience your tasks to fulfil".
4) He treasures up his bright designs
"God moves in a mysterious way" by William Cowper. The tune I usually use is called "London New".
5) He was little, weak and helpless
"Once in Royal David's City" by Cecil F Alexander. Standard Christmas stuff, nothing too demanding.
6) No ear may hear his coming
"O little town of Bethlehem" by Phillips Brooks. We use the tune "Forest Green", but there are at least four other tunes popularly used with these words, one of which was turned into a Christmas single by Cliff Richard. We have been known to sing each verse to a different tune.
7) That mourns in lonely exile here
"O come, o come, Emmanuel" was translated into English by John M Neale from Latin sources dating from between the 9th and 12th centuries.
8) I trace the rainbow through the rain
"O love that wilt not let me go" by George Mattheson. I used this at Anne's funeral and quote it all over the place, particularly the verse from which this line is drawn: "O Joy that seekest me through pain, I cannot close my heart to thee; I trace the rainbow through the rain, and feel the promise is not vain that morn shall tearless be."
9) Heaven's morning breaks, and Earth's vain shadows flee
"Abide with me" by Henry F Lyte. Or the Dead Kennedys, apparently. Popular at football matches, so I'm told. It was used at the wedding of Queen Elizabeth. QE2 if you're English, QE1 if you're Scottish.
10) He will my shield and portion be
Amazing Grace by John Newton. This was written in response to his own reformation as a former slaver.
Most of these are either to do with Advent, Christmas and Epiphany, or were quoted in the lecture on BDSM resonances in Christian Literature.