I learned that X-ray diffraction happens due to the periodic arrangement of atoms in a crystalline material, so can X-ray diffraction studies be done on liquids and non-crystalline materials?
(I'm going to focus on protein crystallography here)
Yes, in fact Small-angle X-ray Scattering, in which (usually) proteins are not in the crystalline state, is used extensively in biochemistry, molecular biology, etc. to determine the size and shape of biomolecules, and to capture large structural changes for which the atomic-scale picture is not necessary.
Additionally, depending on your definition of "liquid", normal X-ray protein crystallography might still interest you, as the scattering of X-rays through imperfect crystals and amorphous solids is an incredibly ripe field for study (it's the field I study).
Protein crystals are by no means actually periodic, they are dynamic systems that are not-identical from unit cell to unit cell, and experience correlated motions that are by no means captured by small, independent, Gaussian (B-factor/thermal factor) motion.
The fact that proteins in the crystalline state are dynamic, and exhibit correlations within and across unit-cell boundaries leads to "continuous" or "diffuse" scattering, which is the cloudy, loosely structured background radiation that accompanies the Bragg peaks in studies which use detectors sensitive enough to measure it.
This diffuse scattering can (in principle) be analyzed to capture information about the dynamics of the protein under study (rather than just the average electron density). In fact, small-molecule X-ray crystallography has developed a robust understanding of diffuse scattering.
One of the most commonly used models to try to understand diffuse scattering in protein crystallography is called the "Liquid-like motions model", proposed by Caspar and Clarage in 1988, which assumed that correlations in electron density fluctuations decay exponentially with distance (think water molecules -- close water molecules move together, water molecules far apart don't really have any relationship to each other at all). This model does remarkably well for how simple it is, but it is definitely not the full story.
Our understanding of diffuse scattering is still very much in its nascency.
Yes. Even single molecule imaging is possible and will be practical in the near future.
More precisely: at modern x-ray free electron laser facilities it is possible to image single particles such as molecules without the need to crystalize them. Currently, there are still many technical limitations, but rapid progress is being made.
See for example HN Chapman, Annual review of biochemistry 88, 35-58 (2019) for a review. From the abstract (emphasis mine):
X-ray free-electron lasers provide femtosecond-duration pulses of hard X-rays with a peak brightness approximately one billion times greater than is available at synchrotron radiation facilities. One motivation for the development of such X-ray sources was the proposal to obtain structures of macromolecules, macromolecular complexes, and virus particles, without the need for crystallization, through diffraction measurements of single noncrystalline objects. Initial explorations of this idea and of outrunning radiation damage with femtosecond pulses led to the development of serial crystallography and the ability to obtain high-resolution structures of small crystals without the need for cryogenic cooling. This technique allows the understanding of conformational dynamics and enzymatics and the resolution of intermediate states in reactions over timescales of 100 fs to minutes. The promise of more photons per atom recorded in a diffraction pattern than electrons per atom contributing to an electron micrograph may enable diffraction measurements of single molecules, although challenges remain.