I'm going to ignore the salutation, because that wasn't what your question was about.
Both are correct, but have slightly different connotations that are related to time.
"The links don't work." - This implies that the links are never going to work until someone fixes whatever is wrong with them.
"The links are not working." - This implies that the links didn't work when you tested them. They may have worked before and they may work after.
Regarding the level of formality, for example, whether or not you should use contractions, you can be too formal in an email. That will depend on your particular workplace. In some places, being overly formal can be interpreted as your being sarcastic.