Single mode fibre I've got a couple of basic doubts about single-mode optical fibers -
a) Why do single-mode fibers have lower attenuation than multimode fibers?
b) Why do light rays in single-mode fibers travel in a straight line?
c) Why do single-mode graded-index fibers not exist? Is it because the diameter and refractive index of such optical fiber would be too low?
 A: 
Why do single-mode fibers have lower attenuation than multimode fibers?

It's an engineering/economic reason, not physics.
The extremely low absorption loss in single-mode fiber (SMF) is obtained by extreme purity of the glass, and by using wavelengths in the 1330 and 1550 nm bands.
Because the transmission distance of multimode fiber (MMF) is limited by dispersion, there's no need to achieve the same loss as in single mode fiber, so as an economic choice, we don't process the glass to the same level of purity, and we use lower-cost lasers and receivers in the 850 nm band (although some newer systems are using 1330 nm in MMF).

Why do light rays in single-mode fibers travel in a straight line?

Put very briefly, because the index variation produces a waveguiding structure that confines the wave to travel along the axis of the fiber. You can see a similar effect in a sound wave travelling in an organ pipe, or a water wave travelling along a narrow canal.
For a deeper understanding you need to solve the EM wave equation with the boundary conditions of the fiber geometry, which you can see done in many optics and electromagnetics textbooks.

Why do single-mode graded-index fibers not exist?

Again, engineering/economic reasons.
Graded index is used for MMF to equalize (as much as possible) the propagation velocities of the different modes. In SMF this isn't necessary because only one mode is propagating.
But graded index fiber is more expensive to manufacture, so we minimize system cost by using step index profiles for SMF.
In comments you asked,

Could you pls explain what is purity/impurity of glass?

Impurity is the presence of other materials besides silica in the glass. For example, iron ion impurities are the reason that ordinary window glass looks green if you look at it from the edge, and eliminating iron impurities was one of the major hurdles to achieve low absorption in optical fiber.
On the other hand, some "impurities" are necessary --- for example, germanium is used to vary the index of refraction to produce the desired index profile of the fiber; but in that case we call the germaninum a "dopant" instead of an "impurity".
