Learn Polish: Profesor napisał nowy wniosek o grant badawczy. - The professor wrote a new application for a research grant.

You're right that wniosek frequently means "conclusion" (as in the phrase wyciągnąć wnioski - to draw conclusions). However, in formal, bureaucratic, or academic contexts, it translates to an "application", "request", or "motion". Whenever you officially apply for something in Poland, you submit a wniosek.

Why do we use the preposition o to mean "for" in this sentence, instead of na or dla?

The noun wniosek (when meaning an application) strongly pairs with the preposition o plus the accusative case to indicate what you are applying for. This stems directly from its related verb, wnosić o (to petition for / to formally request).

Why is the adjective nowy placed before its noun, but badawczy comes after?

Descriptive adjectives that express an opinion or a variable trait, like nowy (new), go before the noun. However, classifying adjectives like badawczy (research) go after the noun because they establish a strict, objective category. It isn't just a grant that is researching; it is the official category of a "research grant".

Grant is an English loanword. Does it decline normally, and what case is it in here?