It is not possible to select nodes and edges at the same time in yEd.
Aside from that, you can use "Tools" -> "Select Elements" to select all neighbors of one or more selected nodes. To do that, make sure that none of the options on tab "General" are selected and on tab "Nodes" choose "Use These Criteria", "Neighbors of Selected Nodes" for "Select", and set "Maximal Path Distance" to "1".