First: you are both right. Are all of the behaviors you listed relatively developmentally appropriate for the age? Yes. Are they also annoying? Yes.
I think you and your husband need to have a conversation when you are both NOT upset about what your six year old is doing. Is she just starting school right now--or in the first few weeks of going back? The reason I ask is that kids can get pretty whiny and distracted during times of change.
I know this struggle you are experiencing because I too have been around children a LOT more than my husband. This makes for some differing opinions, to say the least. I know what it's like to feel like one's job is to 'grease the wheels' between parent and child.
Here's the thing: if your daughter is 'not always listening', then it's time for you to model for your husband what you want him to do in the situation. I find that, instead of brushing it off (which does neither the child nor spouse any good), I have worked on having my son repeat directions back to me as soon as they are given. Because this works well for him, instead of getting upset with my husband, I've given him a tool/technique for getting through this situation.
When we get into potentially troublesome transitions which we can predict are going to be bumpy, I enlist my husband in a quick strategy session. For example, lately it seems that we've had less-than-cooperative behavior around going out for a snack at the local cafe. This is more about the habit of having the conflict than it is about our son not liking to go there. On Sunday we did some pre-cafe planning about how we would try to finesse this transition, and even with a few unexpected glitches, things worked beautifully and we had a nice time together. What really helped was for us to have had that conversation with each other in private and to get on the same page in advance. This way, I wasn't correcting my husband in front of our son- and thus, he felt respected and part of the process instead of just 'wrong'. This might really help your situation if you find that the conflicts are taking place in front of your daughter.
I'd also take it one step further and assign some natural consequences for less-pleasant behaviors. Whining and dramatic? Sometimes it is really appropriate to send them to their rooms to calm down a bit. Sometimes it's okay to say "You know, you've had five minutes of being upset, and we are sorry that you are upset. Now, it's time to decide how you want to solve the problem/how you are going to make yourself feel better."
If your husband is feeling that she's being selfish, teach him that there are times to ignore it and that there are times to pull the discussion out further. WHY does she want what she wants? Sometimes, kids just want to be heard. I do understand that feeling sometimes of "he/she never learns"... let's face it, when our kids are displaying disagreeable behavior, it's frustrating. I would remind him that she is still learning how to be pleasant company, but don't excuse it away. Keep correcting. You know that discipline, when done well, is about correcting and teaching children the right way, not punishing them. Ask him to keep teaching her through proper guidance, and discuss together what that looks like. Maybe she also needs the house expectations and consequences/options written up in a common area: ("If Sally is feeling upset and is not ready to be friendly, she can choose to .... play in the backyard, go play in her room// If Sally is having trouble following directions, we will ...have her repeat the direction back to the parent first, if she chooses not to follow the direction, she will need to sit quietly in a chair until she is ready to do the task asked of her, etc.) .... this would also reaffirm to your husband that he's 'doing it right' also.
Lastly, your husband might enjoy Bruno Bettelheim's "The Good-Enough Parent." It's a thinky book, but I value the advice as it is more philosophical than a 'do this, don't do that' sort of book. Also to get him hooked into how important he is in your daughter's life: The Fatherstyle Advantage is a good affirmation of the positive role he can play.
I know this is long, but as I said: I'm familiar with this challenge. I do hope some of this helps.