I'm looking to create a Conway's Game of Life-type dynamic image that cycles through these general patterns (below).
Is it possible to create these types with Javascript? Or am I better off looking to a 3D application and cycling through an array of pre-define images?
Looking at your examples, I believe you could easily write Javascript code to generate these images. Personally, I would look to generating an SVG. You can do that without a library, but you may find a library like SVG.js helpful to play with.
I would look at the images as a 3x3 grid of your circles that you effectively randomly turn on or off (according to your desired logic). Then you could easily generate the "connector" lines between them.
After you have your 3x3 grid looking how you want it, you could apply a 45° rotation to the group and you are done.
With the above complete, you could then get the svg content and submit it to a server, save it to a file, etc.
If you have any experience with svg / js, you will likely be able to get this done very quickly.
Good luck to you!