غیرممکن نیست

Yes, 'Gheyr-e momken' uses Arabic roots which often sound more formal or academic in Persian than the 'pure' Persian 'Nashodani'.

It sounds a bit dramatic for keys. Better to use 'شاید پیدا بشه' (Maybe they'll be found).

You would say 'تقریباً غیرممکن است'. Note that we use 'ast' (is) here because we are confirming the impossibility.

It might sound a bit cold. 'شاید یه راهی باشه' (Maybe there's a way) is more emotional.

Yes, logically it means it is possible, but pragmatically it emphasizes that it's very hard.

Absolutely. It's a great way to introduce a solution to a complex problem.

The opposite is 'غیرممکن است' (It is impossible) or 'اصلاً راه ندارد' (There is no way at all).

Yes, it is the Ezafe that links the two words. Without it, the phrase is grammatically broken.

No, 'Momken nist' means 'It is not possible', which is the exact opposite of what you want to say!

Modern poetry, yes. Classical poetry usually uses 'Mahal'.

Just change the intonation: 'غیرممکن نیست؟' (Isn't it not impossible?)

Very often, especially regarding diplomatic breakthroughs.

You can say 'تغییر دادن او غیرممکن نیست' (Changing him is not impossible).

It's considered B2 because it involves understanding double negatives and nuanced rhetoric.