Qfyilyi

Yesterday on Dec 10 2018, NASA announced that Voyager 2 probe has exited the heliosphere - the protective bubble of particles and magnetic fields created by the Sun and entered the interstellar-medium; while Voyager 1 reached the milestone back in 2012.

With Voyager 2 has better and more science instruments, however; assume it travels at a higher speed than its current speed (17km/s) and make it to the Oort cloud before the RTGs decayed (still contactable), is it capable of proving or maybe study the Oort cloud?

The Oort cloud named after the Dutch astronomer Jan Oort, sometimes called the Öpik–Oort cloud, is a theoretical cloud of predominantly icy planetesimals proposed to surround the Sun at distances ranging from 2,000 to 200,000 AU (0.0 to 3.2 ly).

In future exploration, it states the Voyager probes will reach the Oort cloud in about 300 years and it will take about 30,000 years to pass through it. As they are still far far away from reaching the starting point of Oort cloud, are the Voyager probes still considered within The Solar System?

If you placed Voyager 1 in the Oort cloud right now, it'd be difficult to contact it (but maybe not impossible). We can barely communicate with the Voyagers now at ~140 AU using a 70 m DSN antenna.
The DSN can use the VLA (one of the largest radio telescopes on Earth), that may provide enough aperture to receive the Voyagers. Calculations later.

Most of the instruments including all cameras on the Voyagers have been switched off. They've been frozen for decades. There is a chance they still work, but I wouldn't want to bet on it.

Can Voyager detect Oort cloud objects? I don't know. The Oort cloud is really dark (little sunlight), which makes detecting the Oort cloud objects difficult. And the Oort could is not a dense region, so the average distance between objects is large.

I actually see two questions here: "Is Voyager 2 capable of proving the existence of Oort cloud?" and "Are the Voyager probes still considered within The Solar System?"

(Not 100% sure on these, this is my understanding so far:)

The boundary of the Heliosphere is considered to be one of the boundaries of our solar system. The Oort cloud is already located in interstellar space. But some still define it to be part of our solar system since the cloud is still bound by the mass of our sun and planets.

So whether you consider the Voyagers to still be within the "solar system" probably depends on how you define what's part of the system. If you go by the definition "up to and including the heliopause", then the Voyagers have left our solar system.

As for the first question: the Voyager probes will run out of energy a long time before they reach the Oort cloud and we thus won't be able to detect them any more. They thus won't contribute to our understanding of the Oort cloud.

Assuming the voyager probes had enough power in their RTGs, had working instruments and could transmit data back to Earth it's still very unlikely they would be able to prove the existence of the Oort Cloud.

The instruments on the spacecraft are not designed to detect Oort Cloud objects, which are theoretically small, sparse and likely to be pretty dark objects to begin with. You'd have to get pretty lucky to spot any objects at all. Even if you saw one that doesn't prove the existence of the cloud, just that there's an object out there in the area the cloud is theorized to be. To prove the existence of the cloud you'd need to spot a few objects at least, map out their composition, position and track their orbits.