Technically they should. If you believe that oort-sized objects exist in decreasing density all the way out to the edge of Sol's sphere of influence, then they would mix with similar objects of our stellar neighbours.

Do you mean two stars' oort clouds intersecting? They could be colliding and intersect that way, but they wouldn't orbit each other and have separate intersecting clouds. In general, star systems are extremely far from each other and wouldn't mix

Yes, this is one of theories explaining asteroid bombardment phases earth suffered in the past. Another start passes nearly and move the oort objects from their orbits.

Don't think anyone knows for sure. However, there may be debris stretching all the way to Proxima Centauri.

Nope. Two stars close enough to have Oort clouds intersecting will be close enough to be directly orbiting each other. The gravitational weirdness of a binary system will clear out anything in the Oort cloud.

But all is not lost, it’s likely another Oort Cloud will form around the binary system.

There is no doubt this must have happened. The ort cloud is so vast and diffuse. Stars travelling in the same galactic plane are unlikely to because they would have similar speed and trajectories, but starts from different sources, like captured globular clusters or other captured galaxies could easily "exchange" ort matter.

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