Thanks!
Yes indeed 15 (with 14 dof): in the example it uses 5 mixture components and each has a weights, location and scale parameter. The choice of 5 will be different for different curves, it's still a bit of an heuristic (look at error & bid-ask violations as a function of number of components).
What's interesting is that if you assume underlying *returns* to have a Gaussian Mixture distribution, then the underlying price will have a mixture of LogNormals distribution, and so the theoretical option price will be a mixture of B&S prices! Brigo, Mercurio & Rapisarda have this same result but starting with a SDE mixture of the underlying. I think implied vol doesn't need a underlying process, I prefer to limit the context of modelling implied to just implied.
What you said about the tails in indeed an important aspect. The Gaussians distributions decay too fast, and there are various ways to try and address that: more mixture components, different base distribution in the mixture,.. or a utility of wealth view. Deep out the money puts are probably still worth something because it's a small cost in the scheme of things to protect against extreme events. Also I doubt people have a clear view on the low extreme probability as a basis for bidding. So modelling the uncertainly of the tail probability as an additional risk factor might be a nice model elements, and also the utility of getting some extra trading capabilities if you have good downward protection. When I was a market maker we had a policy to alway have a net positive number of options at the low strikes.
..and then there is the resolution of the market: you can't get any lower than [zero bid, 0.05 offer]. This is one of the reason I like optimisation based methods that allow you to use creative cost functions like "anything between bid-ask is fine".
A new thing I'm current working on is a Bayesian/probabilistic extension: fit curves to past data and get a distribution of curve shape parameters. Using that distribution we can then 1) better interpolate when you only have few quotes and 2) get a full distribution of curves that fit, model the uncertainty as well. As you said there is a whole family of curves that fit inside the bid-ask. I've tried to model that distribution before with different curve parameterisations but keeping it arbitrage free has been a challenge. Using a density as a starting point eliminates that model risk. It should end up looking like "Gaussian Process Regression" below -where the dot are quote constraints- but with an empirical curve shape distribution (GPR typically uses a-priori chosen curve families), and curves always being arbitrage free.