Do you mean node resize handles with "gray boxes around the groups"?
If so, these are only exported if you activated setting "Paint Selection Marker" in the PNG Image Export settings. (By default, selection markers are not exported.)
If not, please upload a screenshot of your diagram in yEd and the corresponding PNG export to clarify what you mean. (Please see How to upload files to yEd Q&A? for general instructions. However, instead of using the link button, use the image upload button in the toolbar's second row.)
Yes, there is. Use marquee selection to select all nodes in the group as well as the group node itself. Copy the selection. Pasting will then insert all previously copied nodes as well as edges that connect only copied nodes.