Any market maker pricing options with Black-Scholes won't be a market maker for long.
Black-Scholes is just a customer-facing description of the option (i.e, it provides greeks that everyone can understand). But it isn't used as a starting point.
In practice, MM will back out what the implied volatility is from current prices. Then a stochastic volatility model is calibrated against that.
No - no market maker is using stochastic volatility. (L)SV is only used for exotics.
Market makers use a tricked out Black Scholes where the 'secret sauce' is in how you apply the chain rule when you calculate the greeks.
Black-Scholes is just a customer-facing description of the option (i.e, it provides greeks that everyone can understand). But it isn't used as a starting point.
In practice, MM will back out what the implied volatility is from current prices. Then a stochastic volatility model is calibrated against that.
https://en.wikipedia.org/wiki/Stochastic_volatility