If we were to observe some call price (e.g., 15), and then derived implied volatilities from the BS formula depending on different strike prices but fixed maturity (i.e, maturity = 1, and strike goes from 80 to 140??), would we then see a smile?
Yes. Market prices for various strikes and a given maturity often have higher implied volatilities from the Black-Scholes model away from at-the-money.
It is not accounted for in the Black-Scholes model in the fact that volatility is not a function of strike, so volatility is assumed to be constant across strikes, but the market does not price options that way.
I don't know that a quantitative theory has ever been proven; I've always just assumed that people are willing to pay slightly more for options deep in or out of the money based on their strategy, but I have no evidence to base that theory on.