This is one of those transitional periods in technology, with film and video being shot at the same time, depending on who did the shooting.
Don't forget that several of the old newsreel companies were still in business at the time. Universal, Fox Movietone, and a couple others, shot film (mostly B&W) on lots of locations, and continued into the mid 60s.
The live coverage, including the Oswald assassination, was videotape. But I think it was being recorded off air in NYC.
GM: NBC recorded the live video direct, not off the air. So, too, did WBAP and at least one of the two remote trucks they used to feed the signal back to the studio. The CBS live feed from their own camera was also recorded direct and so did their Dallas affiliate, KRLD. The ABC affiliate did not have a live camera in the police basement when Oswald was shot.
The video recorders at the time were huge. Porta-packs, shooting 1 inch tape didn't come about until the late 60s. Most of the JFK press conference stuff, which I think was recorded either at State Department or maybe even GWU's auditorium, was usually B&W film. They didn't build a press room into the White House until after JFK.
GM: JFK's press conferences were all broadcast live; most were held in the afternoons, not at night like they're done now.
David Wolper, who just died recently, has one of the best known documentaries on the JFK assassination called Four Days in November. I think it's mostly in color, but mostly film. It's been a while since I saw it.
GM: It's all b&w film, including kinescopes of some live video.
The TV network situation at the time was case by case. CBS invested a lot of money in its own color system, but was seen by the FCC as not practical. Once the NBC system was approved, CBS moved slowly towards color. ABC was the company that was financially unable to make the full transition. In his biography, ABC chairman Leonard Goldenson says the transition nearly forced the network into bankruptcy. It wasn't until the success of prime time series like Happy Days when the network was able to pay off the transition. ABC's LA studio was built for color in 1967. The other issue was the local affiliates, who also had to make technical changes for color.