Problem: Geometry lines are visible all the way to the eyebag behind the eye.

The eyebag is actually a "surface" like skin, a surface that extends over the eye socket.
Therefore, the actual inner face (opposite the normal), such as the back of the head, is not visible.
But strangely, the eyebag surface is visible.

This makes selecting vertices around the eye area very difficult.

I've tried three solutions, but they still don't work.
1) Enable Remove Backfaces (yellow box)
2) Disable Tint backfaces (red box)
3) Disable Area Select Visible Geometry Only (selection mode options)

How can I make these overlapping lines invisible?

I would use a visibility node, select what you do/don't want to see, then work on the rest.

The remove back faces is ok for viewport use though it doesn't remove the ability
to select points/edges of the faces it's removed (see attached)

Tint changes the default colour of back facing geometry to see them clearer

Using "shaded modes" viewport options works well with "select visible geometry only" and stops you from
selecting anything that isn't directly visible, no surprises there....but wont work with wireframe modes,
even hidden line invisible when selecting points.


Visibility node keeps what is hidden until needed again and just stops you selecting what is
not visible/hidden no nonsense nice and simple.

Edit:

If you need a quick way of selecting front or back facing primitives to hide or whatever, right
click the "geometry select" icon on the left of the viewport or CTRL+SHIFT+B. (primitives ONLY)

Thank you for your reply, EZiniT!.
I thought it might be a simple solution hidden somewhere in the viewport settings.
But as you said, the Houdini settings don't provide a satisfactory solution.
The Visibility node seems to be the easiest and simplest solution.
I'll just have to use that.