The safety zone isn't to protect against the missile exploding. It is to keep the facility far enough away from the likely targets of incoming warheads that they could differentiate warheads from decoys. (As warheads enter the atmosphere the lighter decoys fall away. You want to be at 90* to the warhead flight path to best see this.) Some site were indeed near residential areas in other parts of the country but those were not protecting suburbs. They were protecting more distant targets: population centres and military facilities.

Is that describing the best spot for the interceptor itself, or the radar guiding the interceptor? (Not necessarily at the same site, and I can imaging reasons why separate sites would be better.)

In the days before perfect network connectivity? Both. There are also time delay issues. You want the sensor (radar) the 'computer' and the launch site all close together so they can coordinate the intercept without communication delays, without long radio links that are likely useless during nuclear war.

Sprint was meant for raw speed. It famously accelerated at like 100g. The shortest intercept is one that is launched perpendicular to the incoming warhead. The interceptor goes out vertical as the enemy warhead is coming in more horizontal on its way to the city 400km down the road from the interceptor base.

It probably depended on the geography. I know in the case of the site in Golden Gate Recreational Area, the (Nike not Sprint) interceptor site is/was near the fort there. The control and radar systems (which have been at least partly brought down) were originally on a hill overlooking the site. I don't know if the control/radar system also controlled other interceptor sites or not.

I grew up literally next door to one about 20 miles west of Philadelphia. Most of the Nike sites are just ruins at this point like the one on Angel Island but there's one just over the Golden Gate from SF that's the most fully restored site in the US and gives tours. https://www.nps.gov/goga/nike-missile-site.htm

They initially had conventional warheads as I recall and they were primarily intended as bomber defense. They'd explode a warhead above a flight of bombers so they'd be hit by a massive shock wave.

Eventually, Nike missiles had a selectable yield nuclear warhead. The first salvo was set to intercept at 90 miles away, with 40 Kiloton of yield. The next was 45 miles away, and 20 Kiloton, and the "Hail Mary" was directly overhead, with 2 Kiloton yield.