Is it grammatically correct to use present perfect and past simple tense in the following sentence?
She has been mad at John for five days, as she couldn't go to Paris because of him.
Yes. There's nothing wrong with that sentence, as those are two separate clauses.
Yes. Being mad is a condition which continues to the present, so the perfect is appropriate. (Not) going to Paris is an event which is over, so the past fits.
... as she hasn't been able to go to Paris (perfect) would be fine, but would have a slightly different meaning: it would treat the possibility of her going to Paris as something which might have happened throughout the five days.
Yes, they look good. It's also worth noting that in spoken or written English the use of various verb forms and sentence structures will serve to make your writing/speech more interesting (just as using a good variety of vocabulary does).
Thanks for the question