Did the Big Bang happen?

[HappySkeptic]

Right so my non-science savvy brain would conclude that we have good evidence that the universe was at one point smaller and a lot hotter than it is now. But like I said, do we know with absolute certainty that this was a result of blowing up from a singularity?

No we don't, but how small do you think it was, as it was just "sitting around"?  You can run physics back in time accurately (until you hit new physics).  You can't have a static hot small universe just sitting there.  It doesn't work.  Such a thing either has to be rapidly expanding, or else it will collapse upon itself.

Also, the current state of the universe imposes conditions on the universe at earlier times.

The universe had to be very tiny - so small that we run into new physics.  It either needed to start via a sudden event, or through evolution of some primordial quantum state.  Then, it had to expand rapidly in order to give the level of uniformity (low entropy) we measure in the early universe.