Quick disclaimer: I'm not a mathematician working on Cellular Automata,
but a hobbyist participating in a Cellular Automata community, so some
nomenclature may differ from that accepted in the mathematics community.
1) I've seen this space colloqually referred to as MAP (presumably since
it maps a 3x3 neighborhood into a future cell state), or more precisely
and if you want to be pedantic, since there are a lot of variants of
cellular automata: 2D Range-1 Moore neighborhood 2-state (non-totalistic)
cellular automata (regular euclidean grid implied, although some people
explore toroidal configurations, nonstandard tilings, or arbitrary
graphs).
Changing any of these parameters results in various interesting rulespaces
that other people have explored quite deeply.
2) This is referred to as INT - Isotropic Non-Totalistic Cellular
Automata. The isotropy means all patterns behave the same regardless of
orientation in space, they can be flipped or rotated without changing
their behavior. This makes the MAP space much more manageable to explore,
but still contains many rich and interesting rules.
There are several online communities that explore cellular automata like
these. The biggest one I know of is
conwaylife.com. It includes a very
useful wiki and web forum where rules, rulespaces, search programs,
discoveries, etc. are discussed.
The standard software for exploring cellular automata rules and patterns
is Golly, which supports both these mentioned rulespaces and many more,
including completely custom ones. Many people in the community write their
own tools that allow things like encoding various problems or searches as
boolean SAT problems, which an off-the-shelf SAT solver can usually solve.
This is usually done to search for patterns or rules that satisfy some
property.