My comments, from an assistant chanter in a GOAA parish since 1976, under 3 presiding priests over these years; two of whom graduated from Holy Cross in the mid-'60's; the current one in the early '90's. These responses also apply to having chanted at weddings celebrated by numerous visiting priests.
Re. Psalm 128: Yes, it is always done. The priest intones the verse; the chanter responds, "Glory to Thee Our God..." (The first priest I worked under would skip it, but it's in all the Greek-English service books.)
"Cup of Salvation:" Both of the priests mentioned above, who graduated in the '60's, omitted it, due to the reason you've mentioned; it is a communion hymn, and this is not communion. My current priest does want it chanted. I've never seen a non-Greek priest do it. (I've always wondered how it crept into Greek practice. One priest guessed that it is remnant from when weddings were conducted within the context of the Divine Liturgy, but that doesn't make sense to me, because it is not done in non-Greek practice.)
The Dance of Isaiah:" Each hymn is chanted once, "Dance O Isaiah...;" "O Holy Martyrs...;" "Glory to Thee O Christ..." The priest chants the first one, the chanter, the other two.)
We do not chant the Troparion to Sts. Constantine and Helen, because we have an organist playing, Here Comes the Bride. Typically, previously, "Axion Estin" or "By the Beauty...," was chanted traditionally, especially in Greece, for the bride as she came down the isle. A few years back, when we didn't have an organist that day, my priest chanted the Sts. C & H Hymn at that point. When I asked him why he did not chant "By the Beauty", he said, even in Greece, morality being what it is today, "By the Beauty" is not appropriate and has been replaced by the Sts. C & H troparion.
After checking the references provide above, knowing you as I do from this forum, you probably don't need this advice, but you should check with the priest well in advance. These are the types of items that a priest will have his own way of doing, let alone, what the metropolitan of the metropolis may require.