@happygirl13,
Sure. I will give my opinion with the disclaimer that is very difficult to give a valid opinion after only hearing one side (of course that didn't stop me the first time.
Let's separate the events you have told us so far.
1. In my opinion (based only on what you have told us) buying a Bat Mitzvah dress for your daughter was very inappropriate. I would be upset if that happened to me. I make it clear to my ex, and to any relationship that my daughter has a perfectly good mother and a perfectly good father and no relationship will change this.
For this new relationship to step into your role was completely inappropriate, and for your ex-husband to allow it was very disrespectful and hurtful.
If this happened to me, I think I would ignore the girlfriend. You have no relationship with her (nor should you) and any interaction with her is meaningless.
I think you should take it up with your ex. He should never have allowed his to happen, and as you are co-parenting, you should tell him how hurtful this was. If this were me, I think I would also talk to my daughter... not in an attacking way. But to explain to this that "this hurt my feelings" I think is appropriate.
2. You already know my opinion of the picture. I think it is very minor (especially compared to the Bat Mitzvah issue). The difference is that the picture relates to your ex-husband's relationship with this woman. It has very little to do with the children.
3. I don't see the big deal about a bra from Victoria's Secret. But then I am a man and my daughter is younger than that. It seems to me that a bra is a bra (but what do I know).
4. I would be horrified if my mother were telling my ex-wife things. I think you should end this relationship with his mother.
5. I have know way to judge the fact they were kissing in front of the kids. It depends completely on the nature and quantity of the kisses and how much the kids were really exposed. You weren't there and you were getting information from someone who may have a motive of causing a scandal. Again, I think you should stop talking to his mother.
6. A new relationship has no responsibility to be nice to the ex-wife. I think often this is a cold (if not hostile) relationship (although I know of exceptions). Your ex-husband has the responsibility to treat you with respect and work together with you to raise your children. This woman doesn't.
I also don't think you have any responsibility to treat this woman well.
Unfortunately that's all the opinions I have right now.