Before communication satellites, how did the networks distribute their programming (especially live shows) to the affiliates across the country? Did they use relay towers, land lines, or maybe something else?
There had been national radio networks in the U.S. since the mid-1920s, but these were linked by telephone lines, which didn’t have the bandwidth needed for TV signals. Thus, the earliest TV distribution, in the late 1940s/early 1950s, used kinescopes, which were nothing more than filming a TV monitor when the show was first performed, then shipping the film out to the distant affiliates. The quality wasn’t very good, but this was a era when people were still entranced by test patterns, so people were easier to satisfy.
In the mid-1940s the networks worked with AT&T to develop coaxial land lines for TV signal distribution. The first coaxial line ran from Philadelphia to New York, which was a major reason why both the Democrats and Republicans held their conventions in Philly in 1948. I believe that a nationwide system of coaxial lines was in place by the mid-1950s. About the same time they also developed microwave relay towers, which linked lines of towers on top of hills for point-to-point communication.
Many shows were done in New York in the early days. The most common way to broadcast to the West Coast was the use of kinescopes. A kinescope was a movie type camera set up on a monitor to film the show. Shows were generally done live, and a kinescope was made to make up for the later broadcast times on the West Coast. They were transmitted by co-axial cable.
Your kinescope hypothesis is incorrect. Kinescope is a video-to-film process, it requires the film to be developed, which cannot be done rapidly enough to do what you suggested.
Some live shows did two live performances, one right after the other, for the different time zones. But this was not a very common practice, AFAIK.
It was very common (at least in Canada) to have someone jump on a plane to deliver canisters of film to various TV -stations.
Later a series of microwave towers were built across the country. The towers were used to transmit pix across country, and by individual stations to send stuff to repeater TV transmitters in remote areas (since your average pair of rabbit ears don’t pick up line-of-sight microwaves very well 
)
I believe these towers are still used today in BC, to get pictures from Vancouver (where most stations are located) out to the hinterlands.
NBC mailed video tape to their affiliates in the 1960’s.
The reels were quite large.