Rabbit Hole states:

"When Rabbit Hole is installed, you may search your stack for another copy of Rabbit Hole and install it by paying its install cost. Shuffle your stack,"

My question is could you use the second copy of Rabbit you install to then search for a third copy (should you have two Core Sets) and install it. So you effectively use a single Rabbit Hole to play all three copies, should they be in your stack of course.

Short answer: Yes!

Slightly longer answer: Because it activates when installed, and specifies you find and install another copy (with payment), yes, you can - although it doesn't make sense to shuffle until you're finished with all three (unless you're a masochist) despite the way it's written!

The player on the right was doing exactly this during Gen Con final in this video (around 4:53)

What do I do if I end up with both in my hand before I install the first one? Can I then install the second one from my hand?

And then, since it's not a "May" action, if I install only one regardless, do I have to shuffle my deck regardless of any actions taken?

Isgrimnur said:

Answer to the first question: No, because Rabbit Hole specified searching your stack for a copy to install.

Second question: No idea.

A strict reading of the card says yes, you have to shuffle. This could be nice if the Corp has an ability that messes with the top few cards of your stack.