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I have learned that all spinning objects will continue spinning even if no force is acting on it, and the tendency to do so is called moment of inertia. But I wonder about the fact that a single point mass in a spinning rigid body changes direction, even though no force is acting on it. How this is possible? This violates Newton's laws. How a point mass changes direction without force?

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It does have a force. It is the centripetal force. Usually, the spinning body is a circular object, so all centripetal forces are balanced out to 0. For each point mass, there is another point mass on the opposite side. It is the symmetry of the system that keeps everything in the same place spinning.

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Well, a rigid body is an idealization. One may for simplicity model the various parts of a "rigid body" as held together by springs. When the body rotates, all the springs stretch a bit outwards. If we consider a single mass part of the body, the spring force acting on it then provides the necessary centripetal force, so that it will perform a circular motion.

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All the atoms that make up the spinning body also carry the angular momentum which was imparted to them when the object was accelerated to said angular momentum. When an object is spun to a certain speed the energy goes into the entire object assuming it is one solid piece.

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There are internal forces between atoms that keep each one spinning around the rotation axis. In fact, if you to spin a flywheel really fast, for example, the interatomic forces may not be strong enough, and the flywheel may tear itself apart.

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Simplify it.

Consider a simple bolas - two equal weights spinning around each other held together by a cord. Think of it up in space far away from anything, just to avoid confusion.

Each weight has linear velocity, and each exerts a centripetal force on the other, which gives an acceleration at right angles to the velocity, so as to bend the path of the other into a circle.

So if you draw an imaginary box around the pair of them, no force crosses the boundary of the box to have any affect on the bolas, but internal to the pair, there are forces, acceleration, velocity, momentum, and energy. The energy and momentum are conserved.

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