Is it because the bending of space that occurs around the Sun creates a force acting on the planet which accelerates it towards the Sun ($F=ma$) while the planet has a tendency to move in a straight line at constant speed (as in Newton's 1st law of motion)?
There is only one force acting on the planet, so what prevents the planet from falling into the inferno?
I'll try to rephrase the question. If a planet only experiences one external force directed toward the Sun, why does it orbit instead of crashing into the Sun?
This question falls under the category of "uniform circular motion," so if you want more detail, you can look up that chapter in a physics text. The main idea is the following:
The force of gravity is directed toward the sun. From Newton's law, the acceleration is $\vec{a}=\vec{F}/m$. The acceleration vector is apparently also directed toward the sun, but this does not mean that the velocity vector $\vec{v}$ needs to point at the Sun. So if the velocity starts out pointing some other direction, its direction changes due to the acceleration, but before it has had the time to rotate all the way to the sun, the planet has moved over, so that the velocity vector still isn't pointing at the sun. In uniform circular motion, the magnitude of the velocity (the speed) doesn't change at all, just the direction. In this case, $\vec{v}$ always points tangent to the orbit trajectory and $\vec{a}$ always points toward the center of the orbit (the Sun).
You mentioned the third law as well, so I'll try to clarify that too. The third law says that for any force, there is an equal and oppositely directed force on the other object. In this case, the force we're interested in is the force of gravity exerted by the Sun on the planet. The corresponding third law pair force is the force of gravity exerted by the planet on the Sun. Then think about Newton's second law ($F=ma$) to convince yourself why the sun doesn't particularly care about this force. (Hint: same magnitude of force for Earth and Sun, but different masses).
Planets do not move in fixed orbits, they move in very finely balanced stable orbits.
What you see in the sky today is the result of 4 billion years of thumping, bumping, gravitational games and a lot of stuff falling into the Sun. It's all a very delicate balance of velocity, vector and gravity and it's not all that difficult to upset. While Jupiter won't be leaving the system for anything less than another stellar mass, if you pass a Mars-sized object past our planet it will disturb the orbit enough that our current environmental concerns will quickly become irrelevant.
And don't forget that we have only been studying the solar system for a very, very tiny part of it's existence. We missed all the fun stuff early on.
