Op amp circuit calculations
How is the output impedance derived?

For the purpose of the output impedance of a resistor divider, the two resistors are in parallel.  10 k&Omega; // 10 k&Omega; = 5 k&Omega;.

The following does not agree with the 40 microfarad outcome

You are right.  I must have punched a wrong button on my calculator and not noticed at the time.  I have edited the answer you linked to.

What can be done to reduce the stabilisation time?

With the existing topology, stabilization time is related to gain.  You can reduce the gain, possibly splitting that gain stage into two stages with lower gain.  Or, use a different topology altogether.

Chosing the right topology to best meet the requirements is a important part of designing circuits.