Are all parts of the universe expanding at the same speed?

The expansion of the universe is a funny thing because it isn't expanding like a regular explosion.

The expansion speed (which isn't even a regular speed of movement) of a part of the universe containing another galaxy depends on how far away from you that galaxy is. The farther away from you, the faster it is expanding away from you. So, the speed of expansion of different parts of the universe varies with the part's distance from you.

HOWEVER, this relationship between expansion speed and distance from you can still be expressed as a number (called the Hubble Constant) and that is:
"67.15 ± 1.2 (km/s)/Mpc. For every million parsecs of distance from the observer, the rate of expansion increases by about 67 kilometers per second." - see the source reference below. (A parsec is 31 trillion kilometers, or 19 trillion miles, or about 3.26 light years. A Mpc is a million parsecs.)
Unfortunately, in a way, nothing is simple about the universe, and the Hubble Constant isn't really a constant - it is slowly getting larger as the universe gets older. How fast it is getting larger isn't really known - yet.

No. The expansion only occurs on very large scales. With galaxies, galaxy clusters and galaxy cluster clusters, there is currently no expansion due to gravity stopping it.

All of "space" around and within everything is expanding and, as far as we can tell, at the same speed.

That rate of expansion is not very fast. An atom on the bottom of the foot of a six foot tall person would be moving away from an atom on the top of his head at about 5.43 millionths of an inch per year because of the expansion of the universe.

The four known fundamental interactions can overcome the expansion quite easily at this distance and hold you together. At this scale the electromagnetic force dominates. You remain the same size although space around you (and within you) is increasing in size.
For the Earth and even our galaxy, gravity dominates in opposing the universe’s expansion.

But the pull of gravity changes inversely by the SQUARE of the distance, and at extreme astronomical distances is so faint that it cannot overcome the expansion.

The universe isn't actually expanding at all. The idea that it does comes from the original misinterpretation of galactic redshifts as being a doppler effect when they are actually a scattering effect. The galaxies are not generally receding from each other. Their light simply loses energy through it's interaction with the molecular hydrogen that fills intergalactic space (see source).

If space expanded, as well as stretching out the energy of light radially causing distant galaxies to appear redshifted, it would stretch it out transversely causing a reduction in their surface brightnesses. This is not what we observe:
http://www.sci-news.com/astronomy/science-universe-not-expanding-01940.html

Expanding space would also stretch out the light curves of quasars (their oscillation in luminosity). No sign of this either:
http://phys.org/news190027752.html

This documentary may enlighten you:
https://www.youtube.com/watch?v=IFFl9S39CTM

There never was a Big Bang.

We think local groups of galaxies are not expanding, but some experts disagree.

The Hubble constant is about 70km/s/Mpc. This means objects that are 1 million parsecs (about 3.25 million light years) apart are moving away from each other at a rate of 70 km/s, or rather the space between them is expanding at that rate. Objects 2 million parsecs apart are expanding at a rate of 140 km/s and so on.

great question