This is a normal reaction of somebody who is themselves unhappy. If you can put yourself in your ex's shoes for a moment, she will be suffering too. Her life has also been turned upside down. So, to convince herself that she's made the right decision she has decided to view you as a person with no redeeming qualities and to externalize (My American spell checker keeps putting a z where there should be an s) her pain onto you. But know this... She does not get to decide who you are, you do. My ex tried all this stuff with me and it worked for a long time, until I realized what she was doing. It's not a mature or healthy way to deal with problems.
Don't walk away from your little girl, she needs you. But make it clear to your ex that there is nothing to be gained from talking to you like you don't matter, especially in front of your daughter. Aside from your feelings, it's a terrible example. Keep it civil, if she starts angling for a fight don't take the bait. A good technique is this: Tell her what she did (spoke to you like crap), how it makes you feel (depressed, anxious etc) and what you expect (that she speaks to you with respect).
About the only two things you can't change in this situation are the past and what other people think of you, so don't try. The past is just a story we tell ourselves for how we react in the present. That is the only power it has over us. You know that you have suffered from depression and how that has effected your life, you also know that you are taking positive steps to change yourself for the better. Give yourself credit for that. Those people she says are laughing at you? They only have her side of the story and what does it matter? What kind of person laughs at a man going through a hard time?
You are doing great. There will be hard times but you can get through this. Look at where you are now compared to where you were a few weeks ago.
Practical stuff: Your possessions, if there's anything that is really important, get that and keep it with you or store it with your mother, maybe? The rest? It's only stuff, screw it. Sell it or give it away or let her deal with it. When we broke up, my ex was very aggressive in saying she chose all the stuff in the house to make it look nice and I didn't care. I cared enough to pay for it all, but I didn't want it. It was just stuff. I left with my clothes, my books, my records and my bikes. That was it. And I was much happier.
Anything else she's going back on, get what you originally agreed in writing. Even if its a text or whatsapp. Then hold her to it. Same goes for you, don't agree to anything that doesn't work for you by text or e-mail. You're going to have to be a little hard-nosed about this.
Keep going, you're doing great.
Great post mate.