1) What creates a design trend? One person? A large company re-branding?
Graphic design trends or any design use the same basic process as with fashion trends. There's a life cycle to it and it's usually set up by the biggest players in the market, the design firms with big budgets.
The work is usually done by a small group of designers who work on the new innovative techniques that they actually paid for (no filters, for example). Then it's released in the public, and the trend will expand in the next 5 years. In the end of the cycle, it's usually when you see all the copycats and a new trends are also thrown in the market.
This works for the design style, finish, colors, animation style, etc.
Some examples of trend setters: Madonna and the music industry, Super Bowl ads, TV series intro, fashion, etc.
2) Should you always try and stick with the newest trend? Or can you stick with the old things?
Trends are changing but there's always the classical elements that, as their name says, are more classical and timeless. Yes you can follow trends and anyway you might be expected to on some occasions. But that doesn't stop you from creating your own or sticking to the classical styles; the way you'll decide which one to use is based on that 5 years trend cycle.
For example, you work for a company that has been there for many decades and wants to clearly promote this; then the classical design style suits better. If the company is young and new, or work with a certain target market, then following the trends of the moment is the way to go.
It's an analysis that is based more on your market target than personal choice.
3) What looks like it could be the next design trend? I've seen some low poly art and, in my opinion, it looks like it could be way bigger than it is now.
To know what are the coming trends is to look at music videos from artists like Madonna or any "big machine" artists. And you can also get good clues with popular TV shows in the way they design their intro. You might in fact notice a big pattern when you start looking at big budget designs. You could be right indeed with the poly art if you noticed it a lot on high level promotion.
Usually it's like a pyramid that goes from top to bottom; some big corporation will design something and this will become the next trend that everybody "copy"... The biggest design studios will follow, and then the smaller ones, and finally you'll end up finding these designs in Word template or as free clip part everywhere on the web, for example. So you already know you're not at the top or middle of the pyramid when you use free stuff or when it's easily available as templates!
At least in graphic design, it's a bit more "free for all". In areas like fashion, these trends are selected by a very small group of Creators and the following designers are really expected to stick to them. They will suggest 2 main trends and then everything can be created based on these 2 trends. So in a way it's a bit like a contest where a few elements are selected by Creators, and the designers will play around these themes! You'll rarely see new trends in low price shops; when you do, it usually mean the trend will "die" within the next 1-2 years and is at the end of its cycle.
One interesting point though is the trendy colors for pretty much anything are set by the fashion industry. When the Creators decides aqua is the right color for the next cycle, you'll see aqua used on toothbrushes, computers parts, garden gloves, cars, etc. So if you want to know what are the color trends, the fashion industry is truly the one setting the.. tone!