The First Discourse about the Awakening (Tree)

1: Bodhivaggo The chapter titles in the Udāna are principally named after one (or, as here, more) of the main discourses that occur in the chapter; the only exception to this being the 7^th chapter Cullavagga, the Chapter (including) the Short Discourses. (cont.)

Book, chapter, and discourse titles in Pāḷi are normally quoted in the nominative, which is the so-called label-use of the nominative; see Perniola, PG § 245d. But note that in the manuscript editions of the texts the end-title is the rule, quoting the title at the beginning of a text is a modern felicity.
The Chapter (including the Discourses) about the Awakening (Tree)

Namo tassa Bhagavato Arahato Sammāsambuddhassa
Reverence to him, the Fortunate One, the Worthy One, the Perfect Sambuddha

1-1: Paṭhamabodhisuttaṁ (1) Bodhi in the title here and in the next 2 discourses refers to the Tree of Awakening, not to the Awakening itself, as the events described herein took place some time after that momentous event. Compare 1.4 Nigrodhasuttaṁ (The Discourse about the Banyan Tree).
The First Discourse about the Awakening (Tree)

Evaṁ me sutaṁ:
Thus I heard:

ekaṁ samayaṁ Bhagavā Uruvelāyaṁ viharati, Most of the discourses open with the same formula: Evam me sutaṁ: ekaṁ samayaṁ Bhagavā....viharati. Notice the use of the historical present, literally: ...at one time the Fortunate One...dwells.

The accusative ekaṁ samayaṁ is used with locative sense, another formula which is used just below is: tena samayena, which is the instrumental also used with locative meaning (cf. Kaccāyana on the instrumental § 292: Sattamyatthe ca: sattamyatthe ca tatiyāvibhatti hoti. e.g. tena kālena, tena samayena, yena kālena, yena samayena, tena kho pana samayena).

The Commentary notes that the ancients recognise all these as having locative sense: Porāṇā pana vaṇṇayanti – yasmiṁ samaye ti vā tena samayenā ti vā ekaṁ samayan-ti vā abhilāpamattabhedo esa niddeso, sabbattha bhummam-eva attho ti. Tasmā ekaṁ samayan-ti vutte pi ekasmiṁ samaye ti attho veditabbo.

The locative in the place name which precedes the verb in these formulas is proximate, and usually means near (not at or in), which is more specifically stated in what follows the verb.
at one time the Fortunate One was dwelling near Uruvelā,

najjā Nerañjarāya tīre Bodhirukkhamūle paṭhamābhisambuddho.
on the bank of the river Nerañjarā, at the root of the Awakening tree, in the first (period) after attaining Awakening.

sattāhaṁ ekapallaṅkena nisinno hoti ekapallaṅkena is another example of an instrumental used with locative meaning. The auxillary verb together with the past participle (nisinno hoti) is used in much the same way as the similar construction in the translation: was sitting ; notice that hoti is often used as an auxilliary giving durative sense to the main verb. vimuttisukhapaṭisaṁvedī.
was sitting in one cross-legged posture for seven days experiencing the happiness of freedom.

Atha kho Bhagavā tassa sattāhassa accayena,
Then with the passing of those seven days, the Fortunate One,

tamhā samādhimhā vuṭṭhahitvā, rattiyā paṭhamaṁ yāmaṁ, The Commentary explains rattiyā here as a genitive; and paṭhamaṁ as the accusative of duration: rattiyā ti avayavasambandhe sāmivacanaṁ paṭhaman-ti accantasaṁyogatthe upayogavacanaṁ.
after arising from that concentration, for the first watch of the night,

paṭiccasamuppādaṁ anulomaṁ Anuloma literally: along the hair (or grain); as opposed to paṭiloma (see next discourse), against the hair (or grain). sādhukaṁ manasākāsi:
applied his mind thoroughly to conditional origination in forward order:

“Iti imasmiṁ sati Locative absolute construction, giving conditional sense Syntax § 184b.i. idaṁ hoti; imassuppādā idaṁ uppajjati,
“This being so, that is; from the arising of this, that arises,

yadidaṁ: avijjāpaccayā Paccayā is an ablative functioning as an adverb here, and is really a periphrasis, where the same meaning could have been expressed by the ablative case-ending; the case ending implied in the first half of the compound is the genitive, which is the normal construction when these sort of words (-paccayā, -hetu, -kāraṇā) occur as post-positions. The same construction occurs in Sanskrit, see Syntax § 122, d.

The Commentary takes sambhavanti which occurs at the end of the passage as applying to all the terms but it seems to me that this part of the formula corresponds to the first part of the statement in brief, made above: This being so, that is, being parallel to: with ignorance as condition there are (volitional) processes; with honti being understood. saṅkhārā,
that is to say: with ignorance as condition there are (volitional) processes,

saṅkhārapaccayā viññāṇaṁ,
with (volitional) processes as condition: consciousness,

viññāṇapaccayā nāmarūpaṁ,
with consciousness as condition: mind and bodily form,

nāmarūpapaccayā saḷāyatanaṁ,
with mind and bodily form as condition: the six sense spheres,

saḷāyatanapaccayā phasso,
with the six sense spheres as condition: contact,

phassapaccayā vedanā,
with contact as condition: feeling,

vedanāpaccayā taṇhā,
with feeling as condition: craving,

taṇhāpaccayā upādānaṁ,
with craving as condition: attachment,

upādānapaccayā bhavo,
with attachment as condition: continuation,

bhavapaccayā jāti,
with continuation as condition: birth,

jātipaccayā jarāmaraṇaṁ,
with birth as condition: old age, death,

sokaparidevadukkhadomanassupāyāsā The elements in this compound are grouped and ordered according to the waxing syllable principle, so we have to understand soka-parideva - (2, 4) as one group; and - dukkha-domanass-upāyāsa (2, 3, 4) as a second group. sambhavanti,
grief, lamentation, pain, sorrow, and despair (all) arise,

evam-etassa kevalassa dukkhakkhandhassa samudayo hotī” ti.
and so there is an origination of this whole mass of suffering.”

Atha kho Bhagavā, etam-atthaṁ viditvā,
Then the Fortunate One, having understood the significance of it,

tāyaṁ velāyaṁ Locative absolute, with temporal sense (Syntax, § 183), an idiom which can be matched in the translation. imaṁ udānaṁ udānesi: Udāna, from ud, up, out + √an, to breathe out. The verb is used with the cognate accusative, a common construction in Pāḷi, which cannot be reproduced in English which normally avoids such internal repetition.
on that occasion uttered this exalted utterance:

“Yadā have pātubhavanti dhammā
“When (the nature of) things becomes really manifest

Ātāpino jhāyato brāhmaṇassa,
To the ardent meditating brāhmaṇa,

Athassa kaṅkhā vapayanti sabbā,
Then all his doubts disappear,

Yato pajānāti sahetudhamman”-ti. We might have expected a plural form here in agreement with dhammā in the first line. In the next verse paccayānaṁ in the corresponding line is plural.
Since he knows (the nature of a) thing and its cause.”