As to whether space probes are controlled, or follow a predefined program, it's yes to both.
One way to think about trajectory planning is that space probes are like billiard balls. By being very accurate with the initial hit (in spaceflight: the rocket flinging the probe away from Earth), the future course of the billiard ball can be made so it visits several interesting locations as it bounces around (in spaceflight: encounters the gravitational fields of solar system bodies), with no further propulsion needed.
You can, however, bring some rocket propellant along with you. It eats into the mass budget of things you would rather have brought, like scientific instruments. This extra propellant can be used for small adjustments, or bigger stuff like slowing down when arriving at a planet (otherwise you just fly by).
As for how much fuel space probes use to travel, this is hard to answer. For most of the time, they just coast along, using no fuel at all. The only times a spacecraft needs propulsion is when it need to change its velocity. Reading about delta-v budgets would perhaps be enlightening, but really, no meaningful comparison with car consumption can be made.
Finally, yes, solar panels is the most common way to get power onboard. Other options include 1) Just bringing batteries and run on them until they are empty (short missions) and 2) Bringing an RTG, a small piece of radioactive material giving off some heat and electricity.
Keep in mind that propulsion changing the velocity and course of a spacecraft is completely separate from powering equipment, like cameras and computers, onboard.