RE: Are we thinking about the big bang all wrong?
The right answer is that we don't know... :D
We have observations and assumptions on what happened in the first moments of the universe, but there is very little data on these moments. We have what is called standard cosmology which is somehow the minimal model that works (with respect to the observations we have), but there are alternatives. In addition, god said: never divide by 0. Therefore, at t=0, standard cosmology breaks down. There is in fact no t=0 since the big bang is the moment where time starts to be defined.
Could the edge of the Universe (should there even be one) be a vast series of little bangs expanding and melding into our current Universe?
There is nothing that contradicts this idea. It is just less minimal and less simple (which does not make it wrong). But I will destroy your dream: you are not the first proposing this (see multiverse theories) :p
Pff I figured the answer would be along these lines =P
I know the multiverse theory but I feel mine is distinguished in that it includes constant flow of little bangs within our universe right now causing the expansion in all directions which... I don't think i've heard about before?
You can check conformal cyclic cosmology. Similar to yours somehow :)
I was about to tag you but here you are doing justice to the question :D