Learn Hardcore German: Während der Messe erhielt ich einen hilfreichen Hinweis zu einer nachhaltigen Lösung. - During the fair I received a helpful tip about a sustainable solution.

während is a temporal preposition that governs the genitive case. Since Messe is a feminine noun (die Messe), its genitive singular form is der Messe, not die Messe or der Messe in dative.

Why is the verb erhielt placed before the subject ich in this sentence?

German main clauses follow the Verb-second (V2) rule. Because the sentence begins with the adverbial phrase Während der Messe, the finite verb erhielt must occupy the second position; the subject ich then comes in third position.

What’s the difference between using the Präteritum form erhielt and the Perfekt form hat erhalten? Could I also say bekam?

erhielt is the simple past (Präteritum) of erhalten, typical in written or formal German.
hat erhalten is the present perfect (Perfekt), more common in spoken German.
bekam is the Präteritum of bekommen, a more colloquial synonym. You could say bekam or hat bekommen, but erhalten/hat erhalten sounds more formal.

What case is einen hilfreichen Hinweis and why?

That phrase is in the accusative case because erhalten is a transitive verb that takes a direct object. Hinweis is masculine, so its accusative singular is einen Hinweis; the adjective hilfreich follows the same declension pattern.

Why does hilfreichen take the -en ending here instead of -er or -es?

After the indefinite article einen (masculine accusative), adjectives use the weak/mixed declension and take -en. Pattern:
Masculine Accusative = einen + [adjective-en] + [noun]

Why is it zu einer nachhaltigen Lösung? Which case and preposition rule is at work?

The preposition zu always requires the dative case when used with nouns.
Lösung is feminine; its dative singular article is einer.
After einer, the adjective nachhaltig also takes -en in the dative.
So you get zu + einer (fem. Dat.) + nachhaltigen (adj. Dat.) + Lösung.

What’s the nuance between Hinweis zu einer Sache and Hinweis auf etwas? When do you use zu versus auf?

Hinweis auf etwas often means a pointer or reference directed at something specific (e.g. “a pointer to a section”).
Hinweis zu etwas conveys advice or a tip regarding a topic (e.g. “a suggestion for a solution”).
Here, zu emphasizes practical advice about achieving a sustainable solution rather than merely pointing toward it.

What exactly does nachhaltig mean in nachhaltige Lösung?

nachhaltig literally means “lasting” or “sustainable.” In this context it refers to a solution designed to:

conserve resources long-term
avoid depleting materials or energy
minimize negative environmental or societal impact