Does a (finite) planar graph $G$ with minimum degree 3 contain a vertex $v$ for which all vertices of distance r=2 neighbourhood consisting of bounded degree vertices?
If this is true, does it still work for larger r and on bounded genus graphs?
Somewhat motivated by Can the vertices of a planar graph of min degree 3 be covered with edges of average weight ( sum of degrees) at most 14?
I don't see any reason why this should be the case, or why your condition about degree 2 is here. You can take a tree embedded in the plane such that each vertex has countably infinite valence.