A1. The 1974 Graduale Romanum assigns the following texts to the Introit:

• Deus in loco sancto suo
• Domine, refugium factus es nobis
• Timete Dominum, omnes sancti eius

and the Communion:

• Beati mundo corde
• Primum quaerite regnum Dei

On pp. 1177–1194 of the English translation of The Roman Missal: Third Edition are found the following options for the Introit:

• “May the Lord send you help from the holy place and give you support from Sion. May he grant you your hearts’ desire and fulfill every one of your designs (alleluia).” Cf. Ps 20 (19): 3, 5
• “At dawn, O Lord, fill us with your merciful love, and we shall exult and rejoice all our days. Let the favor of the Lord our God be upon us and upon the work of our hands (alleluia).” Cf. Ps 90 (89): 14, 17
• “I will bless you day after day, O Lord, and praise your name for ever and ever, for you are kind to all and compassionate to all your creatures (alleluia).” Cf. Ps 145 (144): 2, 9

and the Communion:

• “Christ loved the Church and handed himself over for her, to present her as a holy and spotless bride for himself (alleluia).” Cf. Eph 5: 25, 27
• “I give you a new commandment, that you love one another as I have loved you, says the Lord (alleluia).” Jn 13:34
• “I will bless the Lord at all times, praise of him is always in my mouth. Taste and see that the Lord is good; blessed the man who seeks refuge in him (alleluia).” Ps 34 (33): 2, 9

A2. At this time one would be hard-pressed to find musical settings of the Missal introits above. The only one I’m aware of that exists is Richard Rice’s Choral Missal 2010 [PDF], which contains SATB antiphons and verses for all six Missal texts above.

There are multiple English settings of the Deus in loco sancto suo. Fr. Weber (Ignatius Press) and Fr. Kelly have set the Missal translation of this text to plainsong, while Richard Rice in his aforementioned Choral Missal 2010 sets this text to SATB; see their solutions for the Seventeenth Sunday in Ordinary Time.

For metrical settings of the same refer to the two WLP collections: Introit Hymns for the Church Year and Hymn Tune Introits: Singing the Sundays of the Liturgical Year; again, see the Seventeenth Sunday in Ordinary Time.

Andrew Motyka has a responsorial setting of the Communion Primum quaerite regnum Dei [PDF].

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While the question does not ask for offertory texts, all of the above collections (except for Fr. Kelly's) include plainsong and choral offertories for the Nuptial Mass. Also, the following links lead to common-meter (8.6.8.6) paraphrases of not only their respective offertory antiphons from the 1974 Graduale Romanum assigned to the Nuptial Mass but also their verses from the 1935 Offertoriale:

• Immittet Angelus
• In te speravi

Use any common-meter hymn tune that is considered festive in your particular environment.

I think giving couples too many options for these items just adds confusion to something that shouldn't be confusing. Wedding couples at the Cathedral get a form to fill out. They have several choices for things like the entrance processional music, but then when it comes to the more liturgical choices, things become more simple:

Entrance Chant
[full text in English and Latin]
Two check boxes labeled Latin and English

Gloria in Excelsis
Three check boxes: Latin, English, or polyphonic version by the choir

If they check English for the propers they get these:
http://apps.illuminarepublications.com/downloads/lcg/marriage.pdf

If they check Latin, they get that.

If a couple is super into liturgical music, I'll meet with them and make sure we get all the settings and everything they want... but for the average couple? They care about the processional and recessional and might have some preferences for the rest.

I have English versions of the complete GR propers for nuptials in three forms -
1. Gregorian melodies as in GR
2. Anglican chant
3. Psalm tones

The language is traditional Anglican liturgical English.

If this interests you, you may send me your address and I'll mail them to you.

I made this english adaptation of the traditional introit "Deus Israel" (Tobias 7: 15; 8:19). Hope this helps someone.

I'm also sharing the integrated GABC here. I tweaked the chant to fit the text ever so slightly. This was my first attempt at editing GABC. Others who are more adept may prefer to do it other ways or tweak the psalm verse translation. There also seems to be a rendering error when I upload the PDF here, but when I print directly from the illuminare score editor everything prints fine.

name: Deus Israel;
office-part: Introitus;
mode: 3;
book: Graduale Romanum, 1961, p. [121];
transcriber: Andrew Hinkley (English Adapt. J. Richardson);
%font: Georgia;
%width: 5;
%fontsize: 18;
%height: 11;
%spacing: vichi;
%%
(c4) MAY(gj) the(j) God(jjj) of(h) Is(h!iwj/kj)ra(j)el(ji..) *(,) join(gh~) you(hji) to(j)-ge(ih~)ther(h.) (;) and(g) may(h) He(g) be(gfh) with(h!iw!jvI'HG'h) you(hg..) (;) Who(gh) was(//ji) mer(jjj)ci(h)ful(h.) (,) to(h) two(hfh) on(hi)-ly(hg~) chil(hg~)-dren:(g.) (:) and(g) now,(hj) O(j) Lord(i/jjg//h!jjh h.f!gwhgh hg..) (;) make(g) them(gh) more(h!iwj) ful(g_)-ly(gh~) (,) bless(ff e!f'h h g_fhvGF'g) You.(e.) T. P.(::) Al(e)-le(f)-lú(gf~)ia.(g.) (,) al(gf~)-le(g_e/f!gwh_g)lú(ghghFD'ewf/ge~)ia.(e.) Ps.(::) Bless(g)-ed(hj) are(j) they(j) that(k') fear(j) the(ih) Lord:(jjj) *(:) who(ig) walk(hj) in(jji) the(hg) ways(h') of(i) Him.(gh..) (::) Gló(g)ri(hj)a(j) Pa(j)tri.(j) (::) E(j) u(jji) o(hg) u(h') a(i) e.(gh..) (::)