are there any mistakes?
поздравляю С Рождеством!
поздравляю С Новым годом!
С новым счастьем!
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There are no mistakes, only one note: "С новым счастьем!" is very rarely used by itself, usually it is used in conjunction with "С Новым Годом!", like this:
"С Новым Годом, с новым счастьем!" ("поздравляю" is frequently omitted, as it is implied by using the conjunction "С")