Is Light intangible to other Light? And how does all the intersecting light exist in space?

Light is an electromagnetic wave and Maxwell's theory of electromagnetism explains very well how electromagnetic waves propagate in space by the mutual concatenation of time-varying electric and magnetic fields described by Maxwell's equations. Thus it is well understood how light travels in time and space.

Furthermore, Maxwell's equations lead to linear wave equations for the electromagnetic fields. This means that different solutions of them, i.e. different light waves, can be superimposed in space without influencing each other. This might explain what you probably meant by "Is light intangible to other light?"

Let's neglect a specific phenomenon known as interference for now, which is important for special cases, not like a normal light bulb or starlight. Let's also ignore gravitational effects.

"Is light intangible to other light?"

Yes. Two, or more, photons can occupy the same place at the same time. Many photons, or electromagnetic (light) waves, can pass right through each other and not affect each other. So light can intersect with no problem. Like PM 2Ring commented, water waves do the same thing.

Sound waves do too. If you are at a party and everyone is talking, the sound waves from their voices are overlapping while traveling to your, and everyone's, ears, but they do not bounce off or destroy each other. It may seem like they really mess each other up, but that's because your ears cannot focus like your eyes. If you had a really high quality directional microphone you could point it at someone in a crowd and hear them much better. (Also there are some buildings with rooms (maybe the ceiling) shaped like an ellipse or ellipsoid that if you stand at one focal point, you can hear someone talking softly at the other focal point, even if the room is full of people taking.

So back to light. The light from your lamp hits all the points on the walls, and from each point some reflects off in all directions and passes through light from other points on the walls to your eyes and fingers or whatever. It travels in straight lines and does not get deflected by other light it passes through.

As the other answers state, both in the classical mathematical modeling of the behavior of light and in the quantum mechanical where classical light is composed of a superposition of photons, there is no interaction of light, to first order.

The italics to emphasize that in the underlying quantum mechanical frame there exists a photon photon interaction/scattering in higher orders ,with a very very small small probability in the visible frequencies.

The diagram is a shorthand for an integral which allows to calculate the probability of photon photon (two incoming squiggles) scattering off each other (two outgoing squiggles) at energies of visible light. The four electromagnetic vertices make the contribution so small , it can be ignored for visible light frequencies.

The diagrams go on to higher orders in a converging series expansion with diminishing contributions from higher orders, because each vertex gives a $(1/137)^{1/2}$ multiplicative contribution to the final value of the diagram, and the above diagram goes to the fourth power, so already the probability of scattering falls by $\sim 10^{-5}$. Higher orders for visible light means diagrams with more vertices with even smaller contribution.

The electromagnetic spectrum has higher energy photons though, up to gamma rays, and the probability of photons scattering goes up with energy, as students of physics will find when they reach a quantum mechanical level course.

There are even proposals for gamma gamma colliders.