According to the Sunnah, which organ is to be washed first during wudoo’ - nose or mouth?

Also, is rinsing the nose compulsory even if it causes difficulty in breathing?

You have to clean the mouth three times and then when you are done, you clean your nostrils three times according to Sunnah.

Yes, it is compulsory because the nose is part of the face so the evidence used to establish that washing the face is obligatory apply to the nose. When one cleans the nostrils he does not have to breathe water into his nose. You take water with your right hand (not too much water) and put it into your nose and do not stop breathing and you see that nothing goes down your nostrils and you still breathe normally. If you do it correctly you should not find any difficult. Try to go to your nearest masjid and ask the Imaam to show you how to do it. [1]

Reference:

[1] Dr S Al-Fawzan, A Summary of Islamic Jurisprudence 1/41