This answer addresses the part of the question: "How did the upper stage orient itself to point at the horizon while it was spinning? Wouldn't the spinning upper stage be very resistant to changes in its orientation?".
This manoeuvre is called precession of the spin axis. You are right that the spin stabilisation does exactly that, it makes the spin axis more stable to small perturbations, however that doesn't mean that its not viable to do such a manoeuvre. Off the top of my head I couldn't say how much more it would cost in propellant though obviously it needs careful phasing from the thruster driving the precession so that it only fires at the right time.
This is a typical precession manoeuvre (not necessarily that of Explorer 1, works like this, picture this concept:
i) a radially mounted thruster that does not point through the Centre of Mass and is firing continually - there will be no precession,
ii) that it is synchronised only to fire when pointing in one direction - now there will be a turning force (though the reaction has to be consistent with gyroscopic behaviour, my memory is rusty on this). As it will only fire for a small part of the spin period the manoeuvre will take longer than for a non spinning body.
Note that the Juno 1 / Explorer 1 mission was also famous because of a misunderstanding about the nature of spin stabilisation at the time. The answer to this question Explorer 1 describes that point in more detail (I just found and added some relevant material) as the satellite tumbled because of energy transfer due to flexible modes.
I accept the foregoing stops short of saying what actually was the design for Explorer 1. An obvious query would be what combination of sensors and control loops was used for the Explorer 1 mission. What I have been trying to describe is the physics of the problem from a blank sheet of paper perspective.