As a lens designer I can say the info you can tell is quite extensive, but it's not so simple as allowing you to count the number of lenses.
Each "aperture flare artifact" is a separate ghost reflection path which traces from near-focus at the aperture to near-focus at the image plane. It's loosely but not directly an indicator of the number of lenses in the system. The potential number of single-bounce and double-bounce paths that could creat these artifacts is something like 2n + 2^2n, where n is the number of elements in the system. The actual number of artifacts depend on how well these ghost paths were controlled (historically not well). Veiling glare..which you can't see in the example image...is a result of the ghost image paths which are far out of focus. The visibility of the artifacts is an indicator of coating performance. The intensity of the "corona" around bright objects is an indicator of coating and surface quality (resulting from forward scatter at each surface).
All of these effects can be modeled and mitigated with modern design tools. For the applications I design for, this stuff has to be minimized to the greatest extent possible. I would not be a happy camper if I put together a prototype and saw these artifacts, but of course photography is a different application.
Oh, and I wouldn't say the lens used to take the photo would have a decentered element at all. The object is off-axis and the flare artifacts are going in the right direction. The elements are probably centered very well.
Decentered elements induce coma, astigmatism, and chromatic aberration to varying degrees depending on the specific orientations. I see no evidence of misalignment of optics in the presented photo. Trust me on this one... you put together enough prototype optics with really tight tolerances and you pick up on those clues pretty quick.
You'd have to precisely center a small bright object in the FOV to identify decenter from looking at ghost reflections, and you can't line up a lens like that on a camera. You need a centering station or similar optical test station to do that. Easier to deduce misalignment from evaluation of off-axis aberrations.
Regards,
Jason