They have positive non-integer dimensions.
Yes.
Traditional geometric figures have dimensions which are integers: 0 for a point, 1 for a line or Mobius strip, 2 for a plane figure or Klein bottle, and 3 for a solid. Fractals have dimensions which are not integers.
Fractals are a special kind of curve. They are space filling curves and have dimensions that are between those of a line (D = 1) and an area (D = 2).
There are infinitely many fractals so no list can exist.
They can be three dimensional, for example, the Menger Sponge. Mathematically, there is no limit to the number of dimensions.
A fractal in a 2-dimensional plane has a dimension between 1 and 2.
By their very nature fractals are infinite in extent.
No. Idealised objects have 0 or 1 dimensions. Day to day objects have 2 or 3 dimensions. More exotic objects have 4 or more integer dimensions and fractals have fractional dimensions.
Pi is a number. There are no fractals of pi.
Crystals are usually not fractals.
But to a mathematician, it is a neat, neat subject area. Why are fractals important? Fractals help us study and understand important scientific concepts, such as the way bacteria grow, patterns in freezing water (snowflakes) and brain waves, for example. Their formulas have made possible many scientific breakthroughs.
Nobody. Fractals are not owned by anyone!