Certified GKH Gardening Expert
I looked at other online photos of the plant and most of them also had little heart stems growing from the petiole. However, I couldn't find a source that elaborated on this growth. So, your little heart fern is growing normally. It should be interesting to see how it grows from here. I don't think it is coming from the leaf but the point where the leaf attaches to the petiole.
Your fern will do better if you can increase the humidity around it. Low humidity and/or improper watering is causing the brown edges. Let the soil begin to dry out between waterings.