No wonder she is so unhappy.

I go to work (sometimes six days a week) so she can stay home with the kids.

Another way of putting that would be, she raises your kids so that you have time to go to work. If you wanted to be the stay at home dad, you should have thought about that dynamic before you married her. Don't make it sound like you're making a sacrifice. I don't care if you've never phrased it out loud to her like this, I'm sure she can sense your attitude.

We take turns doing dishes and taking the trash out even though she's the one who's home all the time and generating the trash and the dirty dishes

Wow. You've got to be kidding. So, basically you act like you are doing her a favor by doing these things? Would you rather just come home and not have to lift a finger? Well I'm sure she would rather be "off work" at a certain time also. But her work is non-stop. You aren't "helping". You're just taking part in managing your household.

Our house is cluttered and kind of messy but not dirty

So unclutter it. Clean it. What's this have to do with your wife and your marriage problems?

I try to help when she asks me and take the kids so she can get stuff done...

She shouldn't have to even ask you.

but it's stupid things that only takes a few seconds to fix

I would have already left you if you referred to my concerns as "stupid".

I'm leaning toward her just being dramatic.

This is why she is unhappy and you can't do anything right. You undermine her concerns and don't take her seriously. Poor woman. That must be awful for her. Her own husband doesn't even respect her.

When I'm home I help put the babies to bed and pick up whatever toys that got left on the ground by the kids.

Again, this is just ongoing maintenance. Not a favor you are doing for her. Don't act like it is.

I tell my wife to make plans with her girl friends but she always gives excuses not to.

First, I hope by "tell" you mean "suggest". Second, she's freakin depressed. No wonder she doesn't feel like going out.

she's not really taking any steps to do what she used to love

Again, depression does that.

doesn't try to keep up her appearance like she used to

You have to be kidding again. No way you can expect her to look pristine when she has two young kids wearing her out all day and a husband who doesn't respect her.

over a year she still has about 15 lbs of baby weight left

So? 15lbs isn't a major weight gain. Her lifestyle has changed dramatically and on top of that she is miserable all the time. You're lucky it's only that. And what do you mean by "left"? She doesn't have to lose that last 15lbs unless she is really unhappy with the way she looks. She might not mind it....I don't know though.

I don't blame her for leaving. Either adjust your attitude or do her a favor and leave her.