AT A LOSS FOR WORDS= If you are at a loss for words, you don't know what to say because you can't find the right words to express it or because you are too shocked to find the words.

FUNNY= Strange.

THE LAST THING I NEED IS...= I don't really need to...

YOU WOULD SAY= This "would" here is not a conditional tense, it expresses volition and in this case it's an equivalent to "you would like to say".
Actually, the verb WILL may express future or volition. When it expresses volition it is not an auxiliary verb, but a modal verb, and the meaning is the same as "want":
- I will paint the house in blue = I'm going to do it some day
- Will you come and help me, please? = Do you want to help me?
- I asked her, but she won't do it = but she doesn't want to do it
When WILL expresses volition, we can use WOULD as the past tense of "will":
- He knew it, but he wouldn't say = He didn't want to say
- They wouldn't help us, so we had to do it alone = The refused to help us
- I knew he would come = I knew he wanted to come
But WOULD may also be a more polite for of WILL:
- Would you come with me? = Would you like to come? Do you, please, want to come?
The sentence in this song: BUT TO HEAR WHAT YOU WOULD SAY is an example of this last WOULD, a polite form of "will" expressing volition.

WORD OF GOD= Jesus. The first line in the gospel after John, introducing Jesus, says:
"In the beginning was the Word, and the Word was with God, and the Word was God."

WOULD YOU POUR DOWN LIKE RAIN?= The "would" in this sentence is a polite form to express volition (see "you would say" above). It is a polite request, so it means "do you, please, want to descend to me like rain?" or simply "please, descend to me like rain?". POUR DOWN is to make a liquid flow down. Here, they are comparing it with heavy rain pouring down from the skies.

YOUR MAJESTY= God's glory.

STILL= Quiet, without moving.

MIDST= Middle.

BEYOND= Further away than...
Beyond the music, beyond the noise = I can't hear the music or the noise anymore, my mind is further away, and I can only perceive you.

ALL THAT I NEED= The only thing I need.

IN THE QUIET= In silence.