Learn Turkish: Ferahlamak için bu ıvır zıvırı atmak durumundayız. - We have to throw these odds and ends away in order to feel relieved.

Ivır zıvır is a fixed idiom meaning "odds and ends" or "junk." While the 'm-' reduplication you are learning is actively used to mean "[Noun] and things like that" (e.g., kitap mitap), Turkish also has many older, set phrases that use different sounds to create an echo or rhyme. Ivır zıvır is one of these fossilized pairs, so you don't apply the 'm-' rule to it.

Why is the accusative suffix only added to the second word (ıvır zıvırı)?

When you have a paired reduplication in Turkish acting as a single noun—like ıvır zıvır or an 'm-' word like kitap mitap—any case endings are typically only added to the second word. Because the junk is the specific direct object of atmak (to throw), the accusative suffix -ı is placed at the very end of the phrase.

How does durumundayız translate to "we have to"? Doesn't durum mean "situation"?

Yes, durum means "situation." However, the structure [Infinitive verb] durumunda olmak literally translates to "to be in the situation of [doing something]." At a B2 level, this is a very common, slightly more formal way to say "to have to" or "to be obliged to." So atmak durumundayız means "we are obligated to throw [it] away," which carries a bit more weight than just saying atmalıyız (we must throw [it] away).

What is the exact nuance of ferahlamak here?