Q1. What are the effects of multiple broadcast centers at the same planet. Pre patch I found an ai planet that had 3 broadcast centers and a lot of culture. If you have 2 does the culture spread twice as fast? 3 broadcast centers does it spread 3X as fast?
I'm not 100% sure on how it adds up, but I do believe they add up. I also don't know if you can get a higher "cap" or if you just reach maximum culture quicker.
AKA: Devs, lets answer some questions here!

Q2. What exactly does the Refinery do? Does it provide more credits or does it provide metal and crystals? It says it refines the traders raw materials but what is the outcome?
Your asteroid mines have a limited amount of "free" resources, easily gotten at and used. After thats gone, you have to use a refinery to take the "raw" material and refine it into usable stuff.
AKA: They let you harvest resources off depleted asteroids.
Q3. Are the effects of Trade centers cumulative? In other words does building one on two neighboring planets augument the benefits of each? Or do they have no effect on each other.
You
have to have multiple trade centers for them to work. Once you build a trade centers, they send out trade ships to other trade centers that bring in cash flow. The longer the route between centers, the more cash you get out of it. You can't trade interstellar, alas, though refinery ships do go interstellar. This vastly weakens the effects of trade agreements, which allows your trade ships to visit "friendly" trade centers in enemy empires.
Q4. Is there anyway to quantify what benefits Trade centers provide? I am wondering how long it takes to recoup the intial credit cost.
You can highlight a given resource, and it will tell you how much of that resource is currently in trade / refinery ships. Not really precise, but helpful. I can't tell you how long it takes for a given trade center to "pay off", but I will say that the more you build, the quicker they do so. And that they
really increase your economy, to a
huge degree.
Q5. Do existing ships upgrade from new technology automatically or do you need to do something? For example, when you research new armor do all your ships suddenly have this new armor?
Yes, all ships are upgraded once you research the technology, no retrofit needed (yet).