There nothing wrong with using any word when speaking, formal or informal it makes no difference. Using the correct word for meaning is what matters, formal or informal doesn't matter unless the situation requires it ie. wedding toast or business meeting. I understand ordeal.

The word "ordeal" should be used when it fits. It is sometimes the exact word that is needed! Newspapers often use it to describe the experiences of tortured and victimized people. For example, one headline reads, "Syrian women recount regime prison ordeal". It is an excellent word to describe the severe experiences some have gone through.

Ordeal is a bit of an older word, but it can still be used in everyday language, although it's rarely used nowadays. Yes, disaster and nightmare are good synonyms for it, you can also use mishap, misfortune, adversity, tribulation, nuisance, hardship etc.

I think "ordeal" is fine, but to me it suggests something long and drawn-out. "We went camping, and the first night it was windy, and the tent blew down. What a disaster." "We went camping, and it rained five days straight. What an ordeal."

It depends on how well read you are. I don't think people really read as much as they used to so a lot of older words that mean specific things are dropping out of use. I would definitely use 'ordeal' in casual or formal speech, and I like the way 'tribulation' looks up there too. You can also say 'trials and tribulations' to refer to an ordeal(s) if you want to be dramatic. 'She had to work hard through her trials and tribulations to get where she's at.' 'This is a disaster!' or 'What a nightmare!' are also things people say, usually when they are being a little dramatic :)