So, first a little primer on group theory. Nothing too advanced. As you discovered here, you encounter group theory all of the time without realizing it. A group is a set of elements together with some rule that tells you how to combine two elements to get a new one. The set and rule have to obey a few properties:
1. Identity: There's an element that you can combine with any other element that doesn't change the second element. In your case, 00 acts like the identity. Combine 00 with any other bitpair and you get the second one back.
2. Inverses: For any element you pick, there's another element that you can combine with it to get the identity as the answer. For example, the inverse of 01 is 01, since when you XOR them together you get 00, which is our identity.
3. Associativity: This just means that if we have multiple chances to apply the rule, we can do so in any way we choose. So 01 XOR 11 XOR 10 can be performed as (01 XOR 11) XOR 10 or 01 XOR (11 XOR 10) and you'll get the same answer. In either case, you get 00.
4. Closure: This just says that when you combine two elements in your set, the answer will also be in the set. In this case, it's saying that XORing two bitstrings will always give you another bitstring.
So what you've noticed is that the set {00, 01, 10, 11} with the operation of XOR is a group. We say that it's a group of order four, since there are four elements in the set. One question you may have is, are there any other groups of order four? The answer is yes! Let's take a look. To do so, first let's talk about Cayley tables.
A Cayley table is a way to record and visualize how elements of a group interact. You put the elements down a column and across a row in the same order, and then fill in a cell in the table based on how its row and column elements combine. One thing that you'll notice about Cayley tables is that they have that "sudoku" property where every element shows up exactly once in each row and each column. (Can you prove why this is using the group definitions?)
Here's the Cayley table that you already made:
 00  01  10  11


00  00  01  10  11


01  01  00  11  10


10  10  11  00  01


11  11  10  01  00


I'm going to keep the elements the same, but come up with a new rule to combine them that will produce a different table.
 00  01  10  11


00  00  01  10  11


01  01  10  11  00


10  10  11  00  01


11  11  00  01  10


The tables look different, but how do we know that they represent different groups and not just rearrangement of the rows/columns of the first table? Well, in the first table every element was its own inverse. You can see this by seeing all of the 00 elements down the main diagonal. In the second table, only two of the elements are their own inverse (00 and 10). The other two are inverses of each other.
The group denoted by the following Cayley table
is called \(Z_2\) (read as "zee two" or "zee mod two") by mathematicians.
Just like whole numbers can be factored uniquely into primes, groups can be factored uniquely into smaller groups. The group of size four that you made the table for can be factored as \(Z_2 \times Z_2\). The second group of four elements that I made a Cayley table for is \(Z_4\)  there's no way to factor it further. The composition of \(Z_2 \times Z_2\) makes it have the "fractaly" substructures that you see in some of those pictures. These are called "subgroups" by mathematicians  basically, smaller groups that live inside of bigger groups.
Bitstrings of length 4 combined with XOR will give you the group \(Z_2 \times Z_2 \times Z_2 \times Z_2\). In general, bitstrings of length \(n\) with XOR will give you the group that is \(n\) copies of \(Z_2\).
\(Z_2 \times Z_2 \times Z_2 \times Z_2\), which is of size 16, has a subgroup of size 8 and a subgroup of size 4 and a subgroup of size 2, all equal to \(Z_2\) times itself the appropriate number of times (and actually, many "copies" of those subgroups).
If you say that each pixel is made up of a 24long bitstring, then the group represented by your fractal Cayley table is \(Z_2 \times Z_2 \times \dots \times Z_2\), with 24 instances of \(Z_2\). There are \(2^{24}\) elements in that group, and for every power of two less than 24, there's a subgroup with that size. Which helps explain the fractal grids within grids picture that you're seeing.
That's enough group theory for now. I'll read more details and observations from your post and try to respond to them at some point.