When one planet passes near another during its trip around the sun, does their gravitational pull is strong enough to disrupt noticeably each other's orbit ?
It does - although the term 'disrupt' may be a bit too strong to describe the effect; personally, I think 'influence' would fit better.
An interesting consequence of such iterations is something called orbital resonance; after long periods of time - and remember that the current estimate for our planet's existence is 4.54 billion years - the ebb and flow of tiny gravitational pulls cause nearby celestial bodies to develop an interlocked behavior. It's a double-edged sword, though; it may de-estabilize a system, or lock it into stability.
Quoting the Wikipedia entry,
Orbital resonances greatly enhance the mutual gravitational influence of the bodies, i.e., their ability to alter or constrain each other's orbits.
Another gravity-related effect (although, as pointed out by Dieudonné, present only on our solar system between bodies that have very close orbits like the Earth-Moon and Sun-Mercury systems) is known as Tidal locking, or captured rotation.
More about orbital resonance on this ASP Conference Series paper: Renu Malhotra, Orbital Resonances and Chaos in the Solar System.
In fact, the planet Neptune was discovered only after differences between the observed and calculated orbit of Uranus became apparent, and astronomers were able to make predictions about the position of the 8th planet that were eventually confirmed via telescopes.
A similar process led to the discovery of the now ex-planet Pluto.
Depends on what you would call noticeable. The perturbations between the planets are quite small and you will only notice them if you either measure the positions of the planets very accurately or over a very long period. So don't expect two planets to suddenly change direction and move towards each other.
These effect are so small because planets don't actually get very near to each other. If they did then their orbits would be very unstable. If any such planet existed then they would have collided or been ejected from the solar system long ago.
Of course, if you want to calculate accurate positions then you will have to take these effects into account.
Indeed it was the perturbations of Uranus's orbit that led to the discovery of Neptune - which was itself a major triumph of the scientific process and the mathematician Le Verrier is often credited with the discovery even though he did not actually make the observational discovery.
Yes it does, especially with the gas giants.
Jupiter's magnetic field is so strong it yanks on almost everything in the Solar System. Life on Earth also relies on the Jovian magnetic field, because if Jupiter's magnetic field was not there, the gravity would feel much weaker.
After the discovery of Uranus, astronomers predicted another gas giant beyond Uranus due to Uranus being yanked on by something large, which led to the discovery of Neptune by Urbain Le Verrier and John Galle in September 1846.
The Saturnian magnetic field primarily affects Uranus and Neptune.