Why does each celestial object spin on its own axis? AFAIK all the celestial objects have a spin motion around its axis. What is the reason for this?  If it must rotate by some theory, what decides it's direction and speed of rotation?
Is there any object that does not rotate about its axis?
 A: I'm not sure if you want an object that doesn't spin at all, or one that somehow doesn't spin on its axis. In the former, any structure large enough (e.g. superclusters of galaxies) that its dynamical time is longer than the age of the universe is effectively not rotating.
For chaotic spinning (often called tumbling), one example is Hyperion, a small moon of Saturn
A: All celestial objects are formed from larger, more diffuse collections of matter (such as a nebula which collapses to form a star). These larger objects typically have some very small net angular momentum (spin). That total angular momentum is conserved, and as the object collapses it causes the rate of spin to accelerate in order to maintain the same degree of angular momentum. It's the same phenomenon as the spinning ice skater pulling in their arms except in this case the amount of contraction is a factor of millions so even though a proto-stellar nebula may not be rotating much the sheer size difference between a star and a nebula will result in the star rotating a considerable amount.
A: In general yes, everything rotates. It is to do with something called angular moment. Gravity is the central force in the Universe, because it is the only one which has a significant pull over large distances. When things collapse under their own gravity in space (i.e. clouds of gas and dust), any small amount of asymmetry in the collapse will be enough start it spinning. Even if it spins by a tiny amount, as it collapses, angular momentum conservation will mean it spins more and more quickly - just like an spinning ice-skater pulling their arms into their body and spinning more quickly. This means that all coherent masses are spinning - e.g. asteroids, neutron stars, galaxies, quasars.
The Universe is a complex place so something may be slowing down (because the gravity of other objects is putting on the brakes) or some things may appear not to be rotating (e.g. the Moon rotates but at the same rate as it goes around the Earth).
Huge clouds of gas and dust tend not to be spinning as a whole because they are expanding to fill the available volume - like a bad smell in room! - and not necessarily gravitational bound together. However they might have little pockets which start are turbulent, collapse under their own gravity, spin and form stars.
A: Objects in orbit tend to lose their spin on their own axis. 
However they do not completely lose their rotation and end up rotating
with  a period that is the same as the orbital period, so that they
face always the same side towards the other body. The best know example is
the Moon that shows always the same side to Earh.
The phenomenon is called tidal locking.
Tidal locking may happen to both the main body, and the smaller one
orbiting it, but is much faster for the smaller one. Double locking is
faster when the two bodies have similar mass. Double locking is
observable for Pluto and Charon.
The angular momentum cannot be lost and is conserved as orbital
momentum.
Of course, a lone (i.e. far from any other body) rotating body cannot
transfer its momentum and will just keep rotating with the same
period.
But can there be bodies that do not spin. I guess not to spin is with
respect to the distant stars. I am no expert, but I do not believe
that any physical law prevents that, even for smaller bodies, given
that the spin of a body could be compensated by opposite spin of the
larger structure it belongs to. It is probably rare, even considering
approximations, i.e. extremely long periods, in particular because
small structures tend to spin much faster than larger ones. But we
have seen that they can be slowed down.
