Does the Earth’s gravity pull on the Sun and what is the escape velocity from the Sun’s gravity?
The Earth pulls on the Sun with exactly the same but opposite force (this is the effect of applying Newton's Second Law of Motion with his Third Law of Motion).
The escape velocity for an object orbiting the Sun at the same distance as the Earth is about 42 km/sec, but the Earth is already moving around the Sun at about 30 km/sec, so if you go through the gory math, something already moving around the Sun like the Earth would only need an extra 16+ km/sec to escape from the Solar System. And you can actually escape with only about 9 km/sec, because that's enough to get you to the orbit of Jupiter, and if you pass Jupiter in just the right way its gravity will pull you around it and provide an increase in speed that sends you out and away. In fact, that's how all the spacecraft that we have sent to planets beyond Jupiter, and the few that have actually left the Solar System, did the job. We just threw them at Jupiter at about 9 km/sec (when they left Earth orbit), and Jupiter did the rest of the work. (This is called a gravitational boost, and is used in reverse to reach Mercury, by using Venus' gravity to change the orbit; but because the speed change required is much larger, you have to go through several passages to get to Mercury. In fact, the hardest place to get to in the entire Universe is the Sun, because you need to get rid of practically all of the 30 km/sec that the Earth is already moving to get there. (For an example of how such passages were used to send the MESSENGER spacecraft to Mercury, see the image at the end of my web page about Mercury, at https://cseligman.com/text/planets/mercury.htm and for a detailed discussion of that image and how such things work see "Orbital Perturbations and Gravitational Assists' at https://cseligman.com/text/physics/gravitationalassists.htm )
Yes, the Earth's gravity does pull on the Sun. Every object with mass exerts a gravitational force on every other object with mass, and this includes the Earth and the Sun. However, the Sun is so much more massive than the Earth that its gravity is much stronger. This means that the Sun's gravity pulls on the Earth much more strongly than the Earth's gravity pulls on the Sun.
The Sun's gravity is so strong that it keeps the Earth in orbit around it. The Earth is constantly falling towards the Sun, but its motion forward is fast enough that it never actually falls in. This is similar to how a ball on a string keeps going around in a circle. The ball is constantly falling towards the ground, but the string keeps pulling it back towards the center of the circle.
The escape velocity from the Sun's gravity is the speed that an object needs to have to escape the Sun's gravitational pull. The escape velocity from the Sun's surface is about 617 kilometers per second (383 miles per second). This means that if an object were to be launched from the surface of the Sun at a speed of 617 kilometers per second or faster, it would escape the Sun's gravity and never return.
Newton's Theory of Gravity states that every object in the universe pulls on every other object. Every object feels this force, so it is a universal force. The force is always attractive; it is always a pull, never a push. Picture the force of gravity as the tension in an imaginary rope between two objects. You have the basic idea correct, in that the Sun does shift very slightly due to the gravitational pull of the Earth. In fact, the Sun-Earth system orbits about a barycenter that is on average about 449 km (279 miles) away from the center of the Sun. Gravity pulls on itself only in the sense that the pull extends to large distances. Just like a large mass less string was used to pull.Just as the Sun exerts a gravitational pull on the planets, the planets exert a gravitational pull on the Sun. Since the Sun is far more massive than the planets, the pull felt by the Sun is rather small. The escape velocity of the Sun is 615 km/sec, which is more than 50 times greater than the average velocity of H at the surface of the Sun. And since all other elements are heavier than they will have lower average velocities, it is clear that these materials cannot escape from the Sun. The speed required to escape the solar system if you were at the earth's distance from the sun is 42.1 km/s, but the actual escape velocity for something in the earth's system is 16.6 km/s, this is because the earth goes fast, so you get a boost by having that speed. The escape speed of the earth at the surface is approximately 11.186 km/s. That means “an object should have a minimum of 11.186 km/s initial velocity to escape from earth's gravity and fly to infinite space.” Ideally, if you can jump with initial velocity 11.186 km/s you can tour outer space. Escape velocity is the minimum speed needed for a free, non-propelled object to escape from the gravitational influence of a massive body. The escape velocity from Earth is about 11.186 km/s (6.951 mi/s; 40,270 km/h; 36,700 ft/s; 25,020 mph; 21,744 km) at the surface. On the surface of the Earth, the escape velocity is about 11.2 km/s, which are approximately 33 times the speed of sound (Mach 33) and several times the muzzle velocity of a rifle bullet (up to 1.7 km/s). However, at 9,000 km altitude in "space", it is slightly less than 7.1 km/s.