1) What creates a design trend? One person? A large company rebranding?
Trends have to start somewhere. They'll either start with something that designers catch and spread, or they'll start with something that clients love and demand. Responsive Web Design is a good example of the former - people want mobile sites, but there are a lot of ways to do that, and RWD is something that designers prefer, designers notice in action, and designers like to implement. On the flip side, I think you'll get laymen who watch Apple ads, buy from their websites, and then want to have their business look like Apple.
An addendum here is that technology drives trends. Web design didn't exist at all, until, you know, the Internet happened. When scientists are able to programmatically control silk worms to create structures, you'll see new trends in structural design, because we just couldn't do that before.
2) Should you always try and stick with the newest trend? Or can you stick with the old things?
Ask Coca-Cola. Then ask Pepsi.
That's an oversimplification - Pepsi keeps enough branding elements around to maintain familiarity between redesigns, and Coca-Cola's overall branding effort has updated over the years, even if its wordmark has been largely the same for a long time. Same story with GE and IBM - the logos may look 'dated' but they build a brand around it with more modern colors, typography, etc. to give the appearance of modernity.
So, to answer the question? It depends. (Ha!) As Matt_2.0 says, your design should be solving a problem. And the thing about design solving problems is that the problem isn't always solved once and for all. If your solution is still the solution 50 years from now, then there's no need to worry about trends. If it isn't, then you can revise. A great example of this is UPS. Saul Bass designed an iconic and seemingly timeless logo for UPS, and brown became synonymous with the brand. But when UPS wanted to get away from packages tied with string, the logo had to change. The change wasn't entirely driven by a desire to be trendy (though some would contend that the resulting gradient-y shield might suggest otherwise).
If you're going to run a trendy brand, then yeah, stick to the latest trends. But always know why you're doing what you're doing; that's what separates designers from...I don't know what word to use there. So, if you see people using big photos in their web sites, don't be content to say "we need photos here too!" and throw in whatever. Figure out why that photography is being done, what the subject material is, how you can adapt the look to meet your company's goals, and execute your own take on it. If you see long shadows as a trendy thing, make sure your gradient's light source justifies it.
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.
As long as nothing new comes to disrupt the established order, trends are cyclical. Plaid shirts are nice, until everyone feels like there's too much clutter on people's shirts, so they'll wear solid colors until they feel like things are too plain, then they'll go back to plaid. Huge plastic glasses gave way to small, sleek, metal frames, but now the pendulum has somewhat swung back. So maybe that will help you.
Keep an eye on Pantone's colors of the year. Keep an eye on what's going on in the fashion world. Keep an eyes on sites like Brand New to see what new identities are coming out. You can try to set your own trends, sure, but if you feel like you're not in a position to be a trend-setter then follow the best in the business and see what they're doing right.