It is nice to see that you are studying Pali. Overall you seem to have a good grasp of what is going on, and it’s good to see that you are not afraid of experimenting with new renderings. Occasionally your grammar is a bit off, and there are a few other things I would have done differently, too. I have commented on those below. Please forgive me if I seem unreasonably nit-picky and fault-finding!
1. vanapatthapariyāyaṃ vo, bhikkhave, (desessāmi)
“Bhikkhus, I shall instruct you on the forest wilderness.”
This is good, especially “forest wilderness”, which I think is more precise than “jungle thickets”. But you have not translated the word pariyāya, which quite literally means “discourse”, at least in this context. And you have left out desessāmi from the Pali. It needs to be included because it belongs to this sentence. So perhaps: “I shall teach you a discourse on the forest wilderness.”
2. Evaṃ me sutaṃ— ekaṃ samayaṃ bhagavā sāvatthiyaṃ viharati jetavane anāthapiṇḍikassa ārāme.
“This may be translated as: Thus have I heard. On one occasion the Blessed One was staying at Sāvatthī, residing in the Jeta’s Grove, Anāthapiṇḍika’s Park.”
This is fine, but if I was going to be pedantic, I might object to translating viharati twice: “staying” and “residing”. But I am not pedantic, so it’s fine!
3. Idha, bhikkhave, bhikkhu aññataraṃ vanapatthaṃ upanissāya viharati.
“Here, bhikkhus, a certain bhikkhu abides, depending on a forest wilderness.”
In this sentence “a certain” should qualify “forest wilderness”, not bhikkhu. Also, when translating this kind of sentence, it is important to consider what words like idha mean. Often idha just introduces an example, and that seems to be the case here. So it could be translated as “Take the case where” or “In a situation where” or perhaps just “If” or “When”.
4. Tassa taṃ vanapatthaṃ upanissāya viharato anupaṭṭhitā ceva sati na upaṭṭhāti
“If, when abiding supported on the forest wilderness, unattended mindfulness is not looked after”
Nice try, but I am not sure if this makes good sense. And good sense is an important criterion for whether a translation is successful. “Unattended mindfulness” is a contradiction in terms, because mindfulness is synonymous with attention; in other words, it’s a bit like saying “unattended attention” or “unaware awareness”. And so another meaning is required. For me, “unestablished mindfulness” works fine.
Further down you say: “The question is: In this environment can I practice and make progress in mindfulness of the body, of feeling, of mental states, and of Dhammas?” I would argue that it is not so much about whether one “can practice and make progress”, but the more pragmatic question of whether one actually “does make progress”. The proof of the pudding and all that.
5. aparikkhīṇā ca āsavā na parikkhayaṃ gacchanti,
“non-exhausted taints are not exhausted”
I like “exhausted” for parikkhaya; it is better than “destroyed”. “Taints” is probably a bit weak for āsavā, which are the root defilements of the mind. I would suggest “corruptions” or “pollutions”.
6. ananuppattañca anuttaraṃ yogakkhemaṃ nānupāpuṇāti.
“not having reached the ultimate rest from exertion, he does not reach it.”
Here you seem to have read the past participle ananuppattañ (“not reached”) as an absolutive, “not having reached”. Ananuppattañ needs to be read as an adjective that qualifies the main noun, yogakkhemaṃ. In other words, it is the “rest from exertion which is unattained” or the “unattained rest from exertion”. So in a literal rendering of the Pali you have a doubling of the idea of not attaining: “he does not attain the unattained, ultimate rest from exertion”, which can then be simplified into standard English.
7. Ye ca kho ime pabbajitena jīvitaparikkhārā samudānetabbā —cīvarapiṇḍapātasenāsanagilānappaccayabhesajjaparikkhārā—te kasirena samudāgacchanti.
“indeed, also, this one’s requisites of monastic life that should be obtained — robes, almsfood, resting place, and medicinal requisites when ill — are trouble to get.”
I am not sure what you are intending with your translations and whether you wish to mirror the Pali as closely as possible. If you wish to mirror the Pali, especially grammatically, I would suggest to translate as follows: “These (ime – nominative plural, pronoun agreeing with jīvitaparikkhārā) requisites of life (jīvitaparikkhārā – nominative plural, main noun, the effective patient of this passive sentence) that must be obtained (samudānetabbā – passive future participle, agreeing with jīvitaparikkhārā like an adjective and making the sentence passive) by one gone forth (pabbajitena – instrumental, the effective agent of the passive sentence) — robes, almsfood, resting place, and medicinal requisites for the ill (cīvarapiṇḍapātasenāsanagilānappaccayabhesajjaparikkhārā – nominative plural, agreeing with jīvitaparikkhārā) — they (te – nominative plural, being the grammatical agent of samudāgacchanti and agreeing with all the preceding nominative plurals) are obtained (samudāgacchanti – main sentence verb, third person present tense) with difficulty (kasirena – adverb, qualifying samudāgacchanti).”
That is, “These requisites of life that must be obtained by one gone forth — robes, almsfood, resting place, and medicinal requisites for the — they are obtained with difficulty.”
Because it is not clear to me exactly what feedback you are expecting, I will stop there. In general I would recommend you to translate very literally in the beginning, until you get a better feel for the language and how it works. Over time you can take a more relaxed approach to translation, precisely because you have done the groundwork.
In any case, good luck and enjoy your studies. We need more proficient translators!