Christmas in the Park lasts until the third week of January, that would be the 19th so you will catch it for sure, and the flash pass for 10 rides costs between $15 and $30 USD depending on the demand of the day.
About La Feria, yes, going on weekends increases the chances of them running everything, although operations tend to be slow.
There's a new small park built on top of a mall in the south-east part of the city, near Xochimilco, called "Kataplum!" that features a Vekoma Family Inverted Coaster, a Spinning Wild Mouse and many other fun rides:
About non-park related things to do, it really depends what you're into. Teotihuacan is a must-visit definitely.
Chapultepec Park is beautiful, and you can visit many atractions there besides La Feria like the Anthropology Museum, the Chapultepec Castle and the lakes:
Near Chapultepec is the luxurious Polanco neighborhood, there you will find Plaza Carso, home of the Soumaya art museum and Acuario Inbursa (the biggest aquarium in Latin America, which is completely underground):
In the west part of the city, also close-ish to Chapultepec is Parque La Mexicana, a relatively new urban park with fantastic views but not really much to do:
Those are the places that come to mind, don't hesitate to ask me for more specific recommendations!