It won't matter for your citizenship application.
If you are stressing about it, you can apply for the amendment, new PR card will not be issued due to that. So no risk for you.
Don't overthink and don't overcomplicate it.
There's no such thing as applying "using an invitation letter ". She is applying based on her own merits. She can visit her friends, you and whoever she wants. She just needs to show her ties to return.
If you are a representative it's a good practice to use a representative's email, not an applicant's one. Besides, IRCC will use a representative's email by default most often.
Full time study doesn't always mean you need to attend classes 8 hours a day. It can be like evenings, weekends, etc. It's okay to combine it with your full time work and no issues if it doesn't require a study permit.