Is everything quantum entangled? So, I am learning about quantum entanglement in my quantum course and had a thought. If the Big Bang is the creation of everything - time included, and this came from a single "source". Would it not follow that all particles in the universe are entangled to each other? And perhaps, we cannot observe this entanglement because we are "inside the system" sort to speak. Are we not all connected? 

I know the last part is cheesy, but the point is serious.
By far the most particles in the visible universe aren't quantum entangled. That's obvious from observation, since if all electron spins e.g. would be entangled, all electrons would flip their spin at the same time, and we couldn't observe varying statistics of electron spins in a sample, resulting in varying degrees of magnetism, e.g. by applying gradually changing exterior magnetic fields.

A second example: All (bosonic) atoms of spin 0 would form a Bose-Einstein condensate not just near absolute zero, since they would be in the same quantum state, hence in the lowest.

The "single source" isn't a single quantum state. In the early phase of the big bang temperatures were very high, allowing for a huge amount of possible quantum states within a small volume. Without warranty you may think of the initial state of the universe as a superposition of all possible quantum states, the wave functions of which collapse relative to an observer, regarding the observer as a particular quantum state.

That way we get to the many world interpretation of quantum theory and a universal wave function. See also a proposed Hartle-Hawking state as a pre-Planck epoch of the universe.