NASA has just released a telescope alignment evaluation image from JWST for the star HD 84406 2MASS J17554042+6551277. It looks like this:

Higher resolution at Wikipedia
To my untrained eye, the star looks like it would look when viewed from a mirror that has a small smudge at the center. There are multiple lines at 45 degree originating from the star (and also couple of fainter ones at left-center and bottom-center of image).
Is this because of the brightness of the star (it is "only" 258 about 2000 light years away)? Or are there some image post-processing steps that have not been applied to the image? Would the star look the same, if, for example, Hubble Space Telescope photographed it?

Any telescope will have diffraction of the light due to the edges between mirror and non-mirror. This sets a fundamental limit to the telescope's resolution, given by the size of its primary mirror. Earth-based telescopes will usually not have this problem, since the atmospheric seeing will dominate, but James Webb is in space, and so is "diffraction-limited".
The diffraction pattern depends on the shape of the mirror, as well as on anything that is in front of the mirror, i.e. the secondary mirror and the arms holding it (called "struts" or "spider"). In general, an edge will result in spikes perpendicular to that edge. A round mirror, such as Hubble's, results in concentric rings, but its secondary mirror is mounted on a plus-shaped spider, and hence point sources observed with Hubble will have plus-shaped spikes, as seen below:

NGC 6397 globular cluster. Credit: NASA/ESA/H. Richer.
James Webb's mirror is made up of 18 hexagonal segments. The edges of the outermost mirrors hence all follow three different directions, which are aligned at 60° to each other. This is the reason for the six brightest spikes.
Additionally, Webb's secondary mirror is mounted on three arms. The two lower arms are actually aligned with the hexagonal pattern (on purpose, I presume), so the diffraction caused by those fall on top of four of the six spikes. But the upper boom is vertical in the images, and thus gives the fainter, horizontal spikes.
You can see the alignment here on this "selfie" (created using a specialized pupil imaging lens inside of the NIRCam instrument that was designed to take images of the primary mirror segments instead of images of space):

Credit: NASA.
Below you see the resulting diffraction pattern from various spiders:

Diffraction is seen for point sources, which will typically mean stars, because they are bright. The brighter a source is, the brighter the spikes will be. If you exposed for long enough, you'd also see a diffraction pattern from the fainter point sources. Almost all the other sources in this image are galaxies. In principle you also see diffraction from extended sources, but as John Doty comments the pattern will be convolved with the surface brightness of the source, smearing out the spikes.
The bright galaxy seen ~edge-on at ~10 o'clock in the JWST image is a very distant one, with a redshift of 0.285 (source: Grant Tremblay's tweet), placing it at a distance of almost 3.8 billion lightyears.
