In my experience, the best way to handle these underhanded remarks is by responding with a question that has just a bit of edge to it. It puts them in the really awkward position of having to explain why they're being a jerk to you.

You: babadeepboopdoop
Emily: wow, are you sure you're ready to do that?
You: Do you think I can't do it?

You: I did good things!
Emily: are you sure your dad shouldn't edit it?
You: Do you think he wasn't being honest with me when he said it was good?

If she doesn't get the hint after the first few times, then I'd say a direct confrontation is totally fine. Follow up with, "you seem to doubt be a lot by suggesting I'm not ready for things or that others need to take a second look at my work even after they already say they're happy with it. The comments aren't welcome and if you're that concerned about my performance, please take it up with my superiors."

You don't have to make yourself small so that she feels big. She's got a lot of problems to sort out and she's putting her job at risk with her unprofessional behaviour.