Question 2 can be addressed computationally by computing the ranks of generic rigidity matrices corresponding to the sequence of graphs.

I'll show below that the 6th 2-extension for one choice of sequence creates a non-isostatic set from an isostatic one (because a $K_{6,6}$ is formed exactly then). I'm not sure how much this changes as we change the sequence of 2-extensions, nor even how much freedom we have in performing these 2-extensions if we want to go from $K_6$ to $K_{7,6}$. I suspect that the key is in the formation of the cycle $K_{6,6}$.

Below I share what I did in Mathematica; perhaps the code will be helpful for further experimentation. It is a bit tedious, so search this page for "G6" if you want to skip to the exciting part.

First, I wrote some ugly code to compute a 4 dimensional rigidity matrix. No doubt this can be improved.

```
(* p is a list of 4 dimensional vectors corresponding to vertex positions,
E is a list of the pairs of vertices {i,j} (with i<j) that are joined by edges *)
RigidityMatrix4[p_, E_] :=
Module[{e = Length[E], nd = Length[p] Length[p[[1]]], px, py, pz, pw},
Table[
px = p[[E[[j, 1]], 1]] - p[[E[[j, 2]], 1]];
py = p[[E[[j, 1]], 2]] - p[[E[[j, 2]], 2]];
pz = p[[E[[j, 1]], 3]] - p[[E[[j, 2]], 3]];
pw = p[[E[[j, 1]], 4]] - p[[E[[j, 2]], 4]];
Insert[Insert[Insert[Insert[
Insert[Insert[Insert[Insert[
Table[0, {nd - 8}], px, 4 E[[j, 1]] - 3], py,
4 E[[j, 1]] - 2], pz, 4 E[[j, 1]] - 1], pw, 4 E[[j, 1]]],
-px, 4 E[[j, 2]] - 3], -py, 4 E[[j, 2]] - 2], -pz,
4 E[[j, 2]] - 1], -pw, 4 E[[j, 2]]], {j, e}]]
```

Here's a function to create the list of edges corresponding to a complete graph:

```
makecompletegraph[l_] := Reap[Do[If[i != j, Sow[{i, j}]], {i, l}, {j, i, l}]][[2, 1]]
```

Now I begin by creating $K_6$ and deleting the edge {5,6}. The output is a list of the edges as pairs of vertices:

```
G = makecompletegraph[6][[1 ;; 14]]
```

{{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2,
6}, {3, 4}, {3, 5}, {3, 6}, {4, 5}, {4, 6}}

I next compute the number of nontrivial infinitesimal motions of a generic embedding of G and the number of redundancies among its edges:

```
R = RigidityMatrix4[RandomReal[{0, 1}, {6, 4}], G];
{MatrixRank[NullSpace[R]] - 10, Length[G] - MatrixRank[R]}
```

{0,0}

Thus G is isostatic.

I now perform the first 2-extension, by deleting the edges {1,2} and {3,4} and connecting a new vertex 7 to vertices 1 through 6, and check that the resulting graph "G1" is isostatic:

```
G1 = Join[Select[G, # != {1, 2} && # != {3, 4} &], Table[{i, 7}, {i, 6}]]
R1 = RigidityMatrix4[RandomReal[{0, 1}, {7, 4}], G1];
{MatrixRank[NullSpace[R1]] - 10, Length[G1] - MatrixRank[R1]}
```

{{1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3,
5}, {3, 6}, {4, 5}, {4, 6}, {1, 7}, {2, 7}, {3, 7}, {4, 7}, {5,
7}, {6, 7}}

{0,0}

In the second step, I remove {1,3} and {2,4} and attach vertex 8 to vertices 1 through 6 to form G2, which is also isostatic:

```
G2 = Join[Select[G1, # != {1, 3} && # != {2, 4} &], Table[{i, 8}, {i, 6}]]
R2 = RigidityMatrix4[RandomReal[{0, 1}, {8, 4}], G2];
{MatrixRank[NullSpace[R2]] - 10, Length[G2] - MatrixRank[R2]}
```

{{1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 5}, {2, 6}, {3, 5}, {3, 6}, {4,
5}, {4, 6}, {1, 7}, {2, 7}, {3, 7}, {4, 7}, {5, 7}, {6, 7}, {1,
8}, {2, 8}, {3, 8}, {4, 8}, {5, 8}, {6, 8}}

{0,0}

Next {1,5} and {2,6} are removed and vertex 9 is attached to vertices 1 through 6 forming the isostatic G3:

```
G3 = Join[Select[G2, # != {1, 4} && # != {2, 5} &], Table[{i, 9}, {i, 6}]]
R3 = RigidityMatrix4[RandomReal[{0, 1}, {9, 4}], G3];
{MatrixRank[NullSpace[R3]] - 10, Length[G3] - MatrixRank[R3]}
```

{{1, 5}, {1, 6}, {2, 3}, {2, 6}, {3, 5}, {3, 6}, {4, 5}, {4, 6}, {1,
7}, {2, 7}, {3, 7}, {4, 7}, {5, 7}, {6, 7}, {1, 8}, {2, 8}, {3,
8}, {4, 8}, {5, 8}, {6, 8}, {1, 9}, {2, 9}, {3, 9}, {4, 9}, {5,
9}, {6, 9}}

{0,0}

{1,5} and {2,6} are removed; vertex 10 is attached, G4 is isostatic:

```
G4 = Join[Select[G3, # != {1, 5} && # != {2, 6} &], Table[{i, 10}, {i, 6}]]
R4 = RigidityMatrix4[RandomReal[{0, 1}, {10, 4}], G4];
{MatrixRank[NullSpace[R4]] - 10, Length[G4] - MatrixRank[R4]}
```

{{1, 6}, {2, 3}, {3, 5}, {3, 6}, {4, 5}, {4, 6}, {1, 7}, {2, 7}, {3,
7}, {4, 7}, {5, 7}, {6, 7}, {1, 8}, {2, 8}, {3, 8}, {4, 8}, {5,
8}, {6, 8}, {1, 9}, {2, 9}, {3, 9}, {4, 9}, {5, 9}, {6, 9}, {1,
10}, {2, 10}, {3, 10}, {4, 10}, {5, 10}, {6, 10}}

{0,0}

{1,6} and {2,3} are removed; vertex 11 is attached, G5 is isostatic:

```
G5 = Join[Select[G4, # != {1, 6} && # != {2, 3} &], Table[{i, 11}, {i, 6}]]
R5 = RigidityMatrix4[RandomReal[{0, 1}, {11, 4}], G5];
{MatrixRank[NullSpace[R5]] - 10, Length[G5] - MatrixRank[R5]}
```

{{3, 5}, {3, 6}, {4, 5}, {4, 6}, {1, 7}, {2, 7}, {3, 7}, {4, 7}, {5,
7}, {6, 7}, {1, 8}, {2, 8}, {3, 8}, {4, 8}, {5, 8}, {6, 8}, {1,
9}, {2, 9}, {3, 9}, {4, 9}, {5, 9}, {6, 9}, {1, 10}, {2, 10}, {3,
10}, {4, 10}, {5, 10}, {6, 10}, {1, 11}, {2, 11}, {3, 11}, {4,
11}, {5, 11}, {6, 11}}

{0,0}

{3,5} and {4,6} are removed and vertex 12 is attached. The resulting graph G6 is **not** isostatic! Of course, this is because a $K_{6,6}$ is formed.

```
G6 = Join[Select[G5, # != {3, 5} && # != {4, 6} &], Table[{i, 12}, {i, 6}]]
R6 = RigidityMatrix4[RandomReal[{0, 1}, {12, 4}], G6];
{MatrixRank[NullSpace[R6]] - 10, Length[G6] - MatrixRank[R6]}
```

{{3, 6}, {4, 5}, {1, 7}, {2, 7}, {3, 7}, {4, 7}, {5, 7}, {6, 7}, {1,
8}, {2, 8}, {3, 8}, {4, 8}, {5, 8}, {6, 8}, {1, 9}, {2, 9}, {3,
9}, {4, 9}, {5, 9}, {6, 9}, {1, 10}, {2, 10}, {3, 10}, {4, 10}, {5,
10}, {6, 10}, {1, 11}, {2, 11}, {3, 11}, {4, 11}, {5, 11}, {6,
11}, {1, 12}, {2, 12}, {3, 12}, {4, 12}, {5, 12}, {6, 12}}

{1,1}

Just for completeness, I performed the last 2-extension; removing {3,6} and {4,5} and attaching vertex 13. You can see this is $K_{7,6}$:

```
G7 = Join[Select[G6, # != {3, 6} && # != {4, 5} &], Table[{i, 13}, {i, 6}]]
R7 = RigidityMatrix4[RandomReal[{0, 1}, {13, 4}], G7];
{MatrixRank[NullSpace[R7]] - 10, Length[G7] - MatrixRank[R7]}
```

{{1, 7}, {2, 7}, {3, 7}, {4, 7}, {5, 7}, {6, 7}, {1, 8}, {2, 8}, {3,
8}, {4, 8}, {5, 8}, {6, 8}, {1, 9}, {2, 9}, {3, 9}, {4, 9}, {5,
9}, {6, 9}, {1, 10}, {2, 10}, {3, 10}, {4, 10}, {5, 10}, {6,
10}, {1, 11}, {2, 11}, {3, 11}, {4, 11}, {5, 11}, {6, 11}, {1,
12}, {2, 12}, {3, 12}, {4, 12}, {5, 12}, {6, 12}, {1, 13}, {2,
13}, {3, 13}, {4, 13}, {5, 13}, {6, 13}}

{2,2}