This is what worked for me when my kids were younger, may not work for you but it's worth a try....

As soon as the tantrum starts, take her gently by the hand and lead her to another room ( i used to use the hallway) and in a very calm voice say "If you want to have a tantrum you can but you have to have it here and come back when you have finished and are in a better mood" and walk off. At first she will come back as soon as you leave so repeat the process until she stays and the tantrum finishes. Don't talk to her other than to tell her where to have the tantrum until she calms down and keep your voice calm, don't yell. It takes a little bit of practice but I can guarantee after a few times all you'll have to say is "go and have your tantrum somewhere else I don't want to see it" and she will go. We no longer have tantrums because the kids reaslise it's not worth it.

As for the rudeness she is displaying....The first time she is rude to you or DH tell her "You don't speak to mum/ Dad/brother/sister (whoever she is rude to) like that, if it happens again I will take ____ (favourite toy, book, whatever she plays with most) away from you. The 2nd time she does it take it away, put it on top of the fridge and give her a time she can have it back. WIth younger kids I would say you can't have it until after lunch or tea, but with my older kids it ranges from 1 day to 1 week depending on what they have done.

The key is patience and perseverence. These tactics will work as long as you are persistant. At first it will appear that she doesn't care about the punishment but don't be discouraged. She will also go through a rebellious stage where she rebells against the punishment, don't stop, this means it's working! SHe is trying to figure out a way around the punishment so you will cave in....Don't cave!

If you need any help pm me if you wish.