How do we know that ʻOumuamua is interstellar and not otherwise? I fail to understand the logic that the extraordinary speed of the object proves it to be of interstellar origin, and that our sun's gravity failed to capture it.
I'll start with the second question first:
Also, do we exactly know about each and every object inside our solar system?
Each and every object? Of course not. That however is irrelevant to the question at hand. The vast majority of objects in our solar system are very, very small. Quite literally as small as dust. But because they're so small, they have essentially zero impact on the orbits of objects that are not as small as dust.
How do we know that ʻOumuamua is interstellar and not otherwise?
Because it's moving so quickly. ʻOumuamua's huge 26 kilometers per second excess means that is not possible that ʻOumuamua originated within the solar system.
I'll start with the two body problem, e.g., the Sun and some other object interacting gravitationally. The Newtonian gravitational interaction between two objects is a conic section: Either a circle, an ellipse, a parabola, or a hyperbola. The total mechanical energy (kinetic energy plus gravitational energy) for objects following a circular or elliptical trajectory is negative. Such objects are gravitationally bound to one another. Hyperbolic trajectories on the other hand have a positive mechanical energy. Parabolic trajectories form the boundary cases, where the total mechanical energy is exactly zero. Objects on parabolic or hyperbolic trajectories are not bound. They visit once and then they're gone. The excess velocity of an object on a parabolic trajectory (zero excess velocity) or a hyperbolic trajectory (positive excess velocity) is a constant of motion in the two body problem.
The N-body problem makes things a bit dicier. Long-period comets are believed to have originated from within the solar system. These comets appear because something perturbed their very long-period orbits. The perturbation make those long-period comets dive well inside Neptune's orbit. A few of those long-period comets appear to be following an unbound trajectory (non-negative excess velocity). A chance close encounter with a planet can make that happen.
This raises the question: Could ʻOumuamua be yet another long-period comet whose orbit has been perturbed to an extent that puts it on a hyperbolic trajectory? This would not be the first case of an object with an over-unity eccentricity. The answer to this question is no. A close encounter with any of the four known giant planets, or even with the conjectured Planet IX, or even multiple encounters, could not have given ʻOumuamua the excess velocity that has been observed for it.
It's not that much about the speed, but about the orbital eccentricity. Wikipedia gives a good explanation:
Based on observations spanning 34 days, ʻOumuamua's orbital eccentricity is 1.20, the highest ever observed. An eccentricity exceeding 1.0 means an object exceeds the Sun's escape velocity, is not bound to the Solar System and may escape to interstellar space. While an eccentricity slightly above 1.0 can be obtained by encounters with planets, as happened with the previous record holder, C/1980 E1, ʻOumuamua's eccentricity is so high that it could not have been obtained through an encounter with any of the planets in the Solar System. Even undiscovered planets in the Solar System, if any should exist, could not account for ʻOumuamua's trajectory nor boost its speed to the observed value. For these reasons, ʻOumuamua can only be of interstellar origin.