While the below might not perfectly match the criteria in your question, I think it should give you a decent insight in to the kinds of examples which are very closely related to the one you give as an example. There are simply too many examples to check to see which satisfy your stated conditions so I'll just give you a foothold on where to begin your search, as well as a slight exposition of aperiodic tilings. I apologise if this is too much of a tangent from your specific question.
The example you give is actually a generating substitution for (one of the representations of) the Penrose tiling - specifically the Robinson triangles prototile set - here is a very good video which outlines some of the methods that these tiles can be fit together to tile the plane (as well as the other representations of the Penrose tiling). The specific property you mention is related to the notion of a substitution tiling. The tiling associated to the particular substitution in your example looks something like this after a few dozen iterations.
You'll notice that the portion of the tiling shown here isn't periodic (and in fact it can be shown that there is no translation which maps the tiling of the plane to itself), and yet each local region can be found infinitely often through the plane, and within a bounded distance of each occurance. We call such a tiling aperiodic.
Substitution tilings (a sub-branch of the broader theory of aperiodic tilings) are very well studied and an extremely active area of research at the moment; bringing together areas of mathematics such as combinatorics, decidability theory (logic), dynamical systems, non-commutative geometry, mathematical physics, and my current focus - algebraic topology.
I'll answer your fourth question first as it's the easiest - Yes there are substitution tilings in higher dimensions, and even in non-Eucliden spaces such as hyperbolic space. A good place to start would be the work of Chaim Goodman-Strauss - especially for such wonderful images as 'dodecafoam'.
The next easiest is your third question - yes there are fractal substitution tilesets for the plane. In fact, you can cut out little fractal Koch snowflake-like tabs on one of the edges of any polygonal tileset and glue it on to the corresponding edge of the tile(s) that that edge can appear next to after substituting. In terms of pictures, there are also of course the famous Rauzy Fractals
although I don't think this quite satisfies your conditions (I could be wrong).
For some solid examples in the plane that you might like to start with, there's
As well as the half-hex tiling, the sphinx tiling, the table tiling, the Ammann-Beenker tiling, the pinwheel tiling, and many more which you can find in the literature by searching in the areas of aperiodic tilings and substitution tilings.