If dark matter only interacts with gravity, why doesn't it all clump together in a single point?
Great question. Observations show that Dark Matter (DM) only noticeably interacts gravitationally, although it's possible that it may interact in other ways "weakly" (e.g. in the 'WIMP' model --- linked). Everything following has no dependence on whether DM interacts purely/only gravitationally, or just predominantly gravitationally --- so I'll treat it as the former case for convenience.
Observable matter in the universe 'clumps' together tremendously: in gas clouds, stars, planets, disks, galaxies, etc. It does this because of electromagnetic (EM) interactions which are able to dissipate energy. If you roll a ball along a flat surface it will slow down and eventually stop (effectively 'clumping' to the ground), because dissipative forces (friction) are able to transfer its kinetic energy away.
On the other hand, imagine you drill a perfect hole, straight through the center of the Earth, and you drop a ball down it. (Assuming the hole and the earth are perfectly symmetrical...) the ball will just continually oscillate back and forth from each side of the earth to the other --- because of conservation of energy. Just like a frictionless pendulum (no rubbing, no air resistance). This is how dark matter interacts, purely gravitationally. Even if there was no hole through the center of the earth, the DM will just pass straight through and continue to oscillate back and forth, always reaching the same initial height. To zeroth order, dark matter can only 'clump' as much as its initial energy (obtained soon after the big-bang) allows. One example of such a 'clump' is a 'Dark Matter Halo' in which galaxies are embedded. DM Halos are (effectively) always larger than the normal (baryonic) matter inside them --- because the normal matter is able to dissipate energy and collapse farther.
Because the dark matter does not interact a lot, there is no mechanism that would slow it down quickly. When a dark matter particle is falling towards some gravitational center, it is speeding up, then it flies through the periapsis and continues away into the distance. Normal matter clumps into planets, because it is slowed down by interactions / collisions. Dark matter does not collide and cannot deposit energy. It stays on elliptical orbits with very large axes and there is no way how to shrink the ellipse. Normal matter can shrink its orbital path by collisions, but not dark matter.
At this point we know a lot more about what dark matter is not, than what it is. It does not interact via the electromagnetic force, and interaction via the strong force is also unlikely. Interaction via the weak force is still an active area of research (See here).
To understand why dark matter does not form clumps, imagine two particles of dust whizzing through space at high speed toward each other. They get close together but just narrowly avoid a head-on collision before going off in different directions. For a moment, when they were very close together, the pull of gravity between the two objects was at its strongest, but the particles were travelling too fast for the small gravitational pull to hold them together.
Now imagine a different scenario where the two dust particles collide head-on (which happens via the electromagnetic force). Now that the two particles have lost energy through heat, the gravitational pull between the particles can keep them held together in a clump. Soon, a third dust particle comes along and collides into this clump of dust, loses its kinetic energy, and becomes bound to the clump as well. As the clump of dust grows, more particles collide with it and it continues to grow larger and larger, eventually into a planet or star.
Dark matter rarely bumps into itself (or other matter), so it is almost always like the first case, rather than the second. Millions of dark matter particles are passing through you right now without hitting anything. Since it is so hard for them to get rid of their kinetic energy, they tend to not get bound up into clumps.