I'd say COMMON but not NORMAL. Meaning, all kids do this, but it is a behavior that needs correction. It won't "go away" over time.
My daughter is the same age. She is very defiant and we work on that all the time. She has a very sensitive "unfair" button, and if that goes off, she looses all emotional stability and goes into a full blown kicking thrashing temper tantrum. I've had to learn to keep my cool, stay calm and deal with it. If its at that point its best to put her in a room alone to cool off. I only go in if she's kicking the wall or something. I wait for the tantrum to end and then go in and address the issue. She has to calm down and apologize before moving on in the day.
I've also learned how to avoid the behavior. I try not to mention things that "might" happen, or will happen later, because she gets obsessed with them, dwelling on the negative and spiraling out of control.
I need to learn to stop her behavior as its building to an out of control state. Then she can be corrected in a state of mind that is capable of understanding. Once they're out of control, you have to wait until they've calmed down before doing anything or its a waste - or even makes it worse.
Its normal for them to act different at someone else's place. This is a testing time of your authority as the parent. You must win every battle or it just gets worse. A strong willed child delights in the challenge of "beating" you. They believe they can outsmart you and will try at every chance. Every time they win, they desire to do it again.
Keep your cool. Find discipline you and your husband are comfortable with. Give one warning, remind them of the rule (we use a rule chart posted on the fridge, she walks to it and points to the rule + picture) and state the punishment. When the rule is broken, restate the rule and go directly into discipline. Never argue about why they should obey with your child. If they ask, tell them you'll be happy to explain why after they've obeyed. It is going to take a while for them to realize you're serious and not going to give in, but they will eventually. Stick with it.
Best wishes!