I'll answer your question by order:
Yes, the brain does flip the image that is on the retina.
You are mixing 2 separate things. When an object is placed closer than the focal plane to a lens, there will be no real image, only an 'imaginary image' that is not inverted. When an object is placed farther apart, the lens make the rays cross like you said and when you are looking at the object, you see a real inverted image, so your eyes invert it again and there is a straight image on your retina, but your mind flips everything he sees (because your eye lens) so you see the inverted real image.
Glasses help your eyes bend the light in the right way, you do not look at the image the glasses would produce if they were a foot away from your face.
I hope it helped! If you have any more questions you are welcome to ask
The easy answer is "no". In lenses, prisms, mirrors, etc., if the direction of a light ray is reversed, it simply follows the reverse path.
However, depending on why you want such a device, it may be possible to accomplish something close to what you want. For example, there are optical metasurface lenses and polarizing devices that respond differently to light of orthogonal polarizations, so that if (e.g.) right circularly polarized light is incident from the left in your drawing, the lens can be rotated 90 degrees and thereby switch from acting as a positive lens to acting as a negative lens. See, for example, "Multifunctional metasurface lens for imaging and Fourier transform".
There are also angularly selective volume holographic lenses that could do something close to what you want, if instead of simply reversing the lens it were okay to tilt it (and if only monochromatic light were used, and if the only function were to converge and diverge as opposed to forming an image).
Best Answer
The diagram is misleading. In ray diagrams it is conventional to depict the lens as a plane perpendicular to the axis of the lens (in 2D, the vertical dotted line). Then rays only change direction when they meet such planes. It is also a convention to use the paraxial approximation so that all rays parallel to the axis are refracted through the forward focal point of the lens.
This illustration depicts the physical lens superposed on the ray diagram. Those are two separate ways of looking at the refraction of light through the lens.
In reality you are correct. Rays passing through the physical lens are refracted at both front and rear faces. For convex lenses, parallel rays further from the optical axis cross that axis before reaching the focal point.