Select a circle to clean and X > Limited Dissolve. Drag the slider up to around 12 Deg. It will remove most of the unwanted (cylinder in your example) points.
It helps to keep an un-booleaned copy of the mesh so you can shrinkwrap the cut version onto the original - to fix the points on the cut edge that moved in / out.
The bevel will only work as far as the two points nearest each other. Which is about zero in your mesh.
Did you try selecting the cylinder section and doing an inset or an extrude? That may not work for this complex example… Obviously you should have joined the eyes too the rest of the head first.
Well… booleand are “famous” for not being suite to some forms especially with lots of edges and the solution is called apropriate modelling… no… no joking… it’s somekind of holy grail in polygon modeling… see for example on this very long living forum and thread: