1) Could you please tell me more exact translation of this word and of the whole sentence?
"Thanks to his/her(?) kind consideration and personal academic advice to his/her own each student"
2) And in which contexts is this word most suitable?
I think considering the contents of the speech, it is something related with
universities, post graduate, or something about a particular professor, etc.
Here is some extra grammatical information:
細やか is actually an "adjectival noun" (形容動詞{けいようどうし}) , which you can learn almost anything about on the Adjectival noun Wikipedia page.
Since the word that comes after it is 配慮{はいりょ}, which is a noun, it conjugates (similar to verbs).
There is one type of adjectival noun in modern usage, with inflections as follows.
[Note: On the Wikipedia page, you can see all the conjugation forms that are copied below.]
未然形 (Irrealis) -daro
連用形 (Adverbial) -daQ
-de
-ni
終止形 (Conclusive) -da
連体形 (Attributive) -na
仮定形 (Hypothetical) -nara
命令形 (Imperative) ×
Thus, it conjugates into the 連体形{れんたいけい} (attributive), which is the form that leads into nouns: 細やかな
細やか denotes "considerate", "discreet", "minute", etc, etc.