Why are we so fascinated with randomness?

Statistically speaking, we're not. The human mind is exquisitely evolved to see patterns in randomness to the point that we see them when they legitimately aren't there.

If you disagree, let's look at advertisements. Advertisers look at current trends to see how they can market a product, spending a lot of money on research. If we look to most commercials nowadays they are the most arbitrary things ever (e.g. Old Spice, Keystone, etc.) Why is this?

“The thing I hate the most about advertising is that it attracts all the bright, creative and ambitious young people, leaving us mainly with the slow and self-obsessed to become our artists.. Modern art is a disaster area. Never in the field of human history has so much been used by so many to say so little.” ― Banksy

So here's the thing - for every ad you've ever seen, there was an artist (or artists) with an idea. That idea may have started out with how to sell that thing, or it may have started out with a pretty vision. Lots of people in advertising have lots of pretty visions, and they often try out their pretty visions on any product that it might work for. This is one way ad agencies build brand: TBWA/Chiat-Day stuff doesn't look much like Ogilvy & Mather stuff. It may seem "random" to you, but the truth is they're simply using different parameters than you are.

I posit that whatever scientific theory of the time is most prominent, a society/culture will act accordingly.

The problem with your position is that scientific theories aren't even complimentary, let alone hierarchical and societies and cultures rarely act with homogeneity. "Randomness" isn't a theory, for example - if I were to list "prominent" scientific theories today, I'd go with "anthropocentric global warming" "evolution" and "special relativity." The first two are kind of diametrically opposed and the third is irrelevant to either... and "culture" doesn't much give a damn the third while the arguments of the first two are not based in science, but in ideology.

Currently, science states that we live in a disconnected, random, and reductionist Universe headed toward entropy - total chaos.

Actually, "science" states that an equation cannot be solved without selecting a reference frame. But we'll come back to that in a minute. As far as "entropy - total chaos" the end point of an open-ended universe is not chaos, but uniformity. Chaos is actually the opposite of entropy.

But what if we shifted toward believing everything is interconnected, interdependent, and headed toward "centropy" (a state of balanced flux)?

Well, the first problem here is that your reference frame requires your system to be closed. The next problem is that if you're going to pick rules to defy, "thermodynamics" is a tough row to hoe. "A state of balanced flux" is actually a violation of the 2nd law of thermodynamics - the one that prohibits perpetual motion machines. And while you can believe what you want, the preponderance of evidence supports the applicability of this law to physics as we know it.

Euclidean geometry provides answers for 2D events. Great, but the Universe isn't flat. Everything has a 3rd dimension.

Euclidean geometry started falling apart in the late 1500s. The shape of the universe is something debated by cosmologists, and generally not applicable to daily life. I know enough to know that the people who argue about the shape of the universe know more math than I do, and would readily admit that my day-to-day life is unaffected by their determinations.

Similarly, current scientific models of reality are based upon the assumption that closed and finite systems exist.

Could you elaborate on this? Because it's not so much an assumption as a verifiable axiom.

See, what I"m seeing here is that you aren't really defining your system. And I think you're misreading a lot of science to come up with conclusions that aren't really supported by the facts on the ground. It's almost like you're saying "wishing will make it so" while denying things that are easily demonstrated. Absent a better hypothesis, regardless of "truth" or not, the working hypothesis wins, no?