I will write down/out <which one should I use in this case? > some setences to make sure if I get the difference between "hope" and "wish" straight or not.
You must use a conditional mood with "wish".
1. I hope you will get better.
< I think you will get better soon.> You don't have to think this. You could be quite pessimistic about the person's prognosis, and still hope that they get better.
I wish you would get better.
< I want you will get better but I am not sure you will be getting better or not.>
2. I hope I will pass the exam
< I think that I will definitely pass the exam, because I think I did my best> As above. You can hope to pass the exam even if you've done no study.
I wish I could/would pass the exam.
< I want to pass the exam, but I am not totally sure if I will pass it or not>
3. I hope you
willhave a Merry Christmas. (You don't need 'will')
< I think you will have a Merry Christmas.> As above.
I wish you would have a Merry Christmas. (This is an exception)
I wish you a Merry Christmas.
< I want you to have a Merry Chirstmas, but I am not totally sure if you will have a great time or not>
4. Just checking in, hope you're good. This is OK colloquially. "Hope you're well" is better.
Just checking in, wish you're good. No, this doesn't work.