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Much confusion here.
The size of the current universe is 93 Billion (not million) light years in diameter. This is the size of the observable universe. There is no reason to think the actual universe is actually far larger.
The so called singularity from which the universe started was a region of extremely low entropy. There is no reason to think this was necessarily a singularity but could have been part of a much larger region. However, the term "started" is probably a misnomer as there is no reason to think that "Time" has any meaning in the context of this start point. Further, the energy of this region was so high that the Higgs field could not be creating mass so the intuitive concepts relating to mass and its behaviours cannot be applied to this epoch.
Inflation infers a high scalar field applied for a very small time (far less than a second) such that a universe was created out of the low entropy region which explains the CMB we see today. It would be fair to say that the epoch of inflation remains contentious but nevertheless the majority of cosmologists support it pending any other reasonable explanation for the CMB we see today.
After inflation was complete, the universe expanded as per the standard Friedmann equations and our understanding of the general relativity, the four forces and their explanation via GFT, as well as the standard model of particle physics and also the laws of thermodynamics. A plethora of observations support these theories which range from 50-100 years in duration. Add the exhaustive study of the CMB and Hubbles constant and the big bang theory post a fraction of a second is virtually irrefutable to anyone who studies the science.
Our current best bet for epoch of reionisation is in a timeframe of 200K to 400K years but certainly based on CMB data complete after 400K years. New telescopes like the SKA will enable us to better understand this region in the future.
Size of the universe at any time can be calculated from Friedman equations. My understanding of the math at 380K gives the size of the Universe at 380K at 85 million light years diameter, but you would have to confirm this with someone who is better at the math than me.
This is a complex subject. I remember attending a lecture series that lasted 10 1 hour sessions in which the math relating to the big bang was studied in detail. I still have my notes somewhere, but I remember that at the end a very large whiteboard was filled with many tiny equations.
Modern physics stopped being intuitive when quantum mechanics was discovered 100 years ago and even less so when Quantum Field Theory was discovered 50 years ago. The human mind is simply not equipped to understand these concepts without the associated math. The greatest intuitive mind the world has ever seen (Einstein) has extreme difficulty intuiting general relativity until someone else walked him through the math involved. Of course he rebelled against quantum mechanics famously stating that "God does not play dice" as QM defines statistic probabilities of the outcome of the collapse of any given wave function which is probably as counter intuitive as you can get.
As I said the big bang, like evolution, is a complex subject, which explains why a significant number of people do not accept it. Alternative explanations are more easily accessable. Personally, I have no problem with this at all. Where I have a problem is with people who claim to accept science and criticize those who do not, yet they have no real understanding of the science involved in any given subject, yet nevertheless make unsupported claims based on nothing more than a crude understanding of the world.
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