UPDATE: Understanding the Liturgical Worship Service: Service of the Sacrament Part V

Since posting this, I found an hymn by Emanuel Cronenwett that would serve as an alternative hymn for the Nunc Dimittis.

Distribution

Explanation: Jesus physically comes to us in, with, and under the forms of bread and wine, thereby forgiving our sins and strengthening our faith.
Sacramental: Hence the name.
Scripture: Matt 26:26-28, Luke 22:19-20, I Cor 11:23-25.

Nunc Dimittis

Historically: This was not originally a part of the Mass, but instead from the Order of Compline, or Nightsong, sung at the close of the day. I believe it was Martin Luther who put it into the Divine Service.
Explanation: The Nunc Dimittis, or "Lord, Now Let Your Servant Go in Peace," is the song Simeon sang after seeing the fulfillment of God's promise in the baby Jesus in the temple. We too can sing these words of praise and thanks, since we too have seen the Lord's salvation in Jesus Christ. What can give Heathen/Gentiles like us truer peace than this?
Sacrificial: We sing God's Word back to Him.
Scripture: Luke 2:29-32
Alternatives: I'm not aware of any contemporary settings, but see below for alternative texts.
Hymns: LSB 937-938. Also the following, by Emanuel Cronenwett, written in 1876 and published in his "Hymns and Poems" of 1926, and updated just enough so no one notices. Since the meter is LM (88 88), I think PUER NOBIS NASCITUR (LSB 344) would be fine.

Now lettest Thou Thy servant, Lord,

Depart, according to Thy Word,

In peace; for lo! mine eyes have seen

Thy rich salvation from all sin.

By grace divine my soul is fed

With living drink and living bread;

Fed like a branch of Thee, the Vine,

Thy life is mine, and I am Thine.

I live, for Jesus lives in me;

I die, O death, to sin and thee!

Thus living, dying, Jesus saith

My soul shall never taste of death.

Lord, I am weak, be Thou my Stay!

Guide my frail feet along the way;

O help me live the life of faith,

So that I die Thy people's death.

When Thou shalt in Thy glory come

To gather all Thine Israel home,

Then shall my body rise again,

And clothe my soul in light. Amen!

OR

Thank the Lord

Explanation: This is an alternative text to the Nunc Dimittis. Certainly receiving Christ's body and blood is something to thank, praise, and glorify God for, and certainly He recalls His promises, as He did to Simeon, but the rich imagery of the Nunc Dimittis is just not to be found in this text, however joyful the musical setting might be.
Sacrificial: We sing our thanks to God.
Scripture: Any psalm of praise.
Alternatives: Nunc Dimittis
Hymns: May God Bestow on Us His Grace (LSB 823-4), O Jesus, Blessed Lord, to Thee (LSB 632), Sent Forth by God's Blessing (642), O Lord, We Praise Thee (LSB 617)

Thanksgiving/Post Communion Collect

Explanation: Just as with every other Christian meal, a prayer is said, thanking God for His gifts of food, drink, life, salvation, and Christ- in this case received all at once!