The extinction of the Tathāgata will be soon.

Or 'the Tathāgata's extinction ...'. The genitive usually relates to the word immediately following it (see Warder p.56), here parinibbāna.

Na ciraṃ, lit. 'not long'.

Victory will be his.

Imassa, the genitive of the idaṃ pronoun, see Warder p.30. These pronouns are usually demonstrative (i.e. ‘this‘) but sometimes personal (i.e. ‘he‘ or ‘she‘), as in this case.

Or ‘he will have victory‘. Because the genitive usually relates to possession, it can often be translated with ‘have‘.

Brahmins are the sons of Brahmā.

The sentence has no verb and thus the verb 'to be', hoti, is understood. Again, sentences such as this one, with no verb or just the verb ‘to be‘, are 'equational', i.e. 'A = B' or 'A is B'. The 'equation' is between words in the same case, usually nominative (unless the equational sentence forms a clause within a longer sentence). In this case we have 'brahmins = sons' or 'brahmins are sons'. The genitive relates to the word immediately following it, i.e. 'sons', thus 'sons of Brahmā'.

They will make an end of suffering.

Dukkhass'antaṃ = dukkhassa + antaṃ. The last a of dukkhassa has been elided due to the following a of antaṃ, see Warder p.214.

Your argument has been disproved.

Te is here the genitive singular 'of you', see Warder p.56. This can only be known from the context, without which this could also be read as: 'The argument has been disproved by you.'

Āropito is a past participle agreeing with its patient vādo. The sentence is passive.

This is the meaning of this saying.

Another equational sentence where the verb 'to be' is missing. Here the equation is between ayaṃ and attho, 'this = meaning' or 'this is the meaning'.

Two words in the genitive: imassa, a pronoun, and bhāsitassa, a past participle used as a noun. In cases such as this, where a pronoun agrees with a noun (in case, number, and gender), it is very likely that the pronoun qualifies the noun, i.e. '(of) this saying'. Whereas in the Pali each word has the genitive marker '-ssa', in English the preposition 'of', marking the genitive, only occurs once.

Do not stand in front of me.

Aṭṭhāsi is aorist, which is the usual tense with mā, see Warder p.31.

Me, genitive. Purato, 'in front of', takes the genitive, see Warder p.58.

He, through a question to me, I will make clear with an explanation.

'He' refers to someone just mentioned, see below.

Maṃ pañhena. Pañhena should perhaps be regarded as an action noun which takes the object maṃ, 'by questioning me', see Warder p.138.

i.e. 'if you ask me a question about that, I will clarify it for you'. It is difficult to make good grammatical sense of this sentence, in part because it is only the latter half of a longer sentence. The full sentence reads: yassa kho pana Ambaṭṭha mayi kaṅkhā vā vimati vā, so maṃ pañhena, ahaṃ veyyākaraṇena sobhissāmī-ti. The first part, before so, is a relative clause (see Warder pp.70-72): 'But Ambaṭṭha of whom there is doubt or uncertainty concerning me ... '. The 'of whom' must relate to so (the correlative pronoun) and thus so must translate as 'he' (rather than 'that'). But I cannot account for why the Pali has so, i.e. the nominative, when it seems that this pronoun should be the patient of sobhissāmi, i.e. 'I will make clear to him'.

PDF of Syntax of the Cases in the Pali Nikayas (opens in new tab)

Now, at that time, Ānanda was standing behind the Blessed One, fanning the Blessed One.

Ṭhito hoti, lit. 'was stood' (historic present tense, thus hoti becomes 'was'). Sometimes the Pali idiom cannot be translated directly into English, therefore 'was standing', or simply 'stood', or maybe 'was stationed' for a more literal rendering. Also note that the word pana is used to 'join' this sentence to the previous one, hence it can be translated as 'but', 'however', 'now', or 'then'.

Piṭṭhito takes the genitive, thus the genitive form Bhagavato. One could perhaps translate piṭṭhito with 'at the back of' to make this relationship more clear.

Vījamāno, present participle agreeing with Ānando. Bhagavantaṃ is the patient of vījamāno.

However, while I am doing the work, (my) body will become tired.

Kammaṃ kho pana me karontassa is an example of the construction called ‘genitive absolute‘, see Warder p.58. A genitive absolute construction requires a participle in the genitive case (here karontassa, present participle genitive) and an agent of the participle also in the genitive (here me, enclitic genitive of ahaṃ, 'I', see Warder p.56). On translation the genitive absolute clause loses its genitive sense but is made to stand apart from the rest of the sentence by introducing it with 'while' or 'when' etc. Note that kammaṃ kho pana is also part of the absolute construction, with kammaṃ being the patient of karontassa, see Warder p.58.

Note that despite the translation, kilamissati is an active verb, lit. '(the body) will tire'.

The gems are his.

Or 'they are his gems'. Another 'equational' sentence. Here ratanāni is 'equated' with the agent of bhavanti, 'they', thus 'they are gems'. The genitive tassa, 'his', relates to the word immediately following it, thus 'his gems'.

Note that although bhavanti is the present tense, the context in the sutta makes it clear that the future is being spoken of. This is an example of what Warder on p.13 calls 'a vivid future visualised as present'.

The context is the prophecising at the birth of the Bodhisattva Vipassī that, if he stays at home, he will become a universal emperor who has 'seven gems/precious things'.