I have recently taken an interest in shadows. I know that in order for a shadow to exist that you must have a solid in the way of the light. My hypothesis is that there can be a light so strong, like a laser beam, that it acts like a solid in the sense where it doesn't let light pass through... is this plausible?
Your hypothesis is basically correct, in theory, even in a vacuum.
Light consists of electromagnetic radiation. According to classical electrodynamics, electromagnetic fields in a vacuum are linear, which means that one light beam will pass right through another, completely unaffected. But according to quantum electrodynamics, at electric field intensities greater than the Schwinger limit, electromagnetic fields are expected to become nonlinear even in a vacuum, resulting in inelastic photon-photon scattering. That more or less means that a beam of light could in principle "block" another beam of light that crosses it.
However, the Schwinger limit at this point is purely a theoretical limit. Even the Extreme Light Infrastructure currently being developed, which will be at the frontier of producing super-intense laser light, will still produce light that's a couple orders of magnitude below the Schwinger limit.
Yes and no.
Photons don't interact in free space. So a beam of light can't block another beam of light in vacuum.
Photons can interact due to the nonlinearity of the medium. So it's plausible to block another beam of light if you have the right mediators.
It's however not the light itself becoming a solid. See, for example, electromagnetically induced opacity.