Learn Turkish: Bağış toplamak şöyle dursun, kirayı ödemek imkânsız. - Let alone collecting donations, paying the rent is impossible.

Literally, it means "let it stand like that" or "let it stay over there" (şöyle = like that, dursun = let it stand, the 3rd person imperative of durmak). It is a fixed phrase used to push aside an extreme or unlikely idea (collecting donations) before pointing out that even a basic task (paying rent) is impossible. You can often swap it with bir yana.

Why is kira marked with the accusative suffix (kirayı) but bağış is not?

Kira takes the definite accusative suffix (-yı) because it refers to a specific, known rent (the rent for the speaker's building). Bağış (donation) is used here as a general, indefinite object, so it stays in its bare dictionary form right before the verb.

Why do we use the dictionary infinitive -mak (toplamak, ödemek) rather than the verbal noun -ma?

When making general statements without a specific grammatical subject attached to the action, Turkish uses the full -mak infinitive. The pattern [Verb]-mAk şöyle dursun is the standard way to express "let alone doing [action]" as a general concept.

Could I swap the order of the actions, like we sometimes do in English?