Cosmological constants I've heard that the cosmological constant is 0.[123 more zeros] and then a 1 [in some units].
Does that means that it used to be exactly zero? Is the value of this constant changing or is it fixed at a particular value at the Big Bang?
 A: I think you may be confusing two "versions" of the cosmological constant.
First there is $\Lambda$ (or sometimes $\lambda$), which expresses the absolute energy density of "dark energy". Depending on convention, this constant can have a variety of dimensions. You seem to have the dimensionless (or Planck unit) version in mind, which is:
$$\lambda\sim10^{-122}$$
or as you say, about 0.0000[continue until you get 121 zeros]1. In the standard $\Lambda$CDM model of cosmology, this values never changes.
The cosmological constant is also often expressed in terms of the critical density $\rho_{\rm crit}$, in this case it is denoted $\Omega_\Lambda$:
$$\Omega_\Lambda=\frac{\Lambda}{\rho_{\rm crit}}$$
Because the critical density changes with time, $\Omega_\Lambda$ also changes with time. Early in the history of the Universe it was nearly zero (the critical density was higher), but has been steadily increasing, and is thought to currently be the dominant form of energy density in the Universe. The present measured value is:
$$\Omega_{\Lambda,0}\approx 0.692\pm0.010$$
A: Does that means that it used to be exactly zero?
No. See Wikipedia, and note this:
"In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: Λ) is the value of the energy density of the vacuum of space".
You will be aware that we have no evidence whatsoever that challenges conservation of energy, and that we have very good evidence that the universe is expanding. These together suggest that the cosmological constant is reducing over time. Don't worry about the issue of constants not being constant. See how Kyle mentioned the fine structure constant? Have a read of what NIST say: "Thus α depends upon the energy at which it is measured, increasing with increasing energy, and is considered an effective or running coupling constant". It's a running constant. It's a constant that isn't constant! Obviously there's something of an issue with what Kyle said about ΛCDM saying Λ never changes, but trust me, that can be fixed. As for "this number changing", don't worry about it. For all you know you know you owe your very existence to this number changing. And as for the Susskind interview, the cosmological constant isn't "a kind of antigravity", the vacuum catastrophe is a problem for the tiny number, and the "Goldilocks" universe is a myth because electron mass varies with gravitational potential (see the mass deficit) and the speed of light isn't constant. If it was, there wouldn't be any gravity. Plus we don't actually know that the universe is very much bigger than the observable universe. That's an inference that's arisen in the last few years since WMAP discovered that the universe was flat. And of course there's no actual evidence for bubble universes or a multiverse or "megaverse". IMHO Susskind is rather indulging in speculation here, and it comes over as fact and/or consensus. If I were you I'd ask questions here about that sort of stuff. 
