Comets from Oort passing Sun, are captured back in Oort's reservoir?

Probably not.

Consider a comet in the Oort cloud, about 50,000 AU from the Sun. At this distance, the comet's potential energy relative to the Sun is very high, almost high enough to escape from the Sun altogether.

Now imagine that something disturbs this comet very slightly; maybe the gravitational tug of a passing star. The comet starts to descend Sunward, constantly converting potential energy to kinetic energy as it speeds up. If the Sun were the only gravitational body, the comet would pass by it very closely with a speed nearly equal to the Sun's escape velocity. But the Sun ISN'T the only gravitational influence; it is surrounded by eight planets, a handful of dwarf planets, asteroids, and other bodies, each of which exerts a tiny tug on the comet. As the comet descends into the warmth of the inner solar system, vents open up that push the comet like rockets, further changing its velocity and orbit. In some scenarios, the comet actually gains enough energy to slingshot past the Sun on a one-way trip into interstellar space. It will not rejoin the Oort cloud; it will just become an interstellar wanderer.

The other outcome is that the comet actually LOSES enough energy that its aphelion is lowered out of the Oort cloud. Such comets may return to the Sun again and again, maybe hundreds or thousands of years apart.

In either case, once a comet is dislodged from its position in the Oort cloud, it's usually gone for good...inward or eventually outward.

I hope that helps. Good luck!

Not very likely... there was some force that knocked that comet out of the cloud & inward toward the sun; once the Comet has made it's loop around the sun and head back out to the cloud, it'll need some force again to keep it there - to change it's orbit. If no force is applied, it will continue in that orbit for a very long time.

Unlikely. The solar system escape velocity at the distance is very low, about 200kph. Mot comets either lose of gain energy when they come into the solar system so they are either ejected or captured. In some very very very very rare incidences their might be just enough change that the gravitational influence of the Oort cloud would be just enough to pull them back round. But is is a long shot.

Very unlikely, but not impossible. It wouldn't take much force to get the comet back into a Oort orbit once it returned to the Oort cloud. Problem is, there isn't much out that far to supply the needed force.