In comments you added,
The wavelength that I use is 633 nm the diameter of the core is 8 micrometer.
IIRC, standard silica-glass 9-um fiber with ~0.5% index difference from core to cladding supports 3 or 4 propagating modes at 633 nm.
This means you have either single-mode fiber or something very close to it, unless the index contrast is much stronger than in standard fiber.
And this means your diagram showing many propagating rays, with dramatically different angles, is at best a gross simplification of the actual behavior, and likely to lead you to wrong conclusions about the physics.
To really understand the fiber behavior, you should consider a wave optics model rather than ray optics.
Do we get in the end of the fiber Evanescent field (Total reflection)?
Presence of an evanescent field does not imply total internal reflection.
There will always be an evanescent field present at a fiber end facet, if the modes profiles in the 2nd medium don't exactly match the mode profiles in the fiber.
There will not be total reflection when a fiber end is coupled to free space.
If we get Evanescent field in the end of the fiber, what is the shape (intensity profile of the field) then?
As far as I know, a complete 3-d EM simulation would be needed to answer this.
I have question about also the traveling of Gaussain beam in the fiber, Does it hit always in the same position on the cladding in total reflection or it can be hit in another spots on the cladding as it shown in the following picture
First, the mode profiles for (low mode-count) step-index fiber are not Gaussian. They are normally described by the Bessel functions. The difference between a Gaussian profile and a Bessel profile are pretty small, but they're enough to not allow perfect coupling from a fiber mode to a single free space mode.
Second, (here's where your ray optics model is leading you astray), the beam doesn't "hit" in specific locations along the cladding boundary. The beam propagates as a guided wave, with (for the main LP01 mode) nonzero energy anywhere in the core, and an evanescent component in the cladding, all along the length of the fiber.