The "no-name Chinese copies" are in many cases really good. I have a Star B92m, which is roughly a Galaxy S III copy. It is slower, and slightly different port layout (which is annoying, as otherwise it is perfect dimensions to fit all cases etc.). Apart from that the only problem is that the GPS antenna is way too short, so if GPS matters to you you pretty much need to open the phone and mod it. Other than that it's an excellent phone.
The B92m is last years model, and I bought it for about $240 at the time. This year you can get quad core full hd (1920x1080) phones for the same price.
Basically it is down to reading reviews in advance to ensure the features you care about works as they should.
Safety is really not a concern when it comes to the phone itself - if you're worried, throw away the charger and pick up any USB charger you trust more.
Overall keep in mind that most of these "copies" are only copies or imitations of better known brands on the surface - if you open them up what you tend to see is boards that are closely following the same reference boards for either MTK (Taiwanese fab-less manufacturer; one of the worlds largest semiconductor companies), Rockchip or AllWinner (Cinese mainland) CPUs.
You can pick up devices from dozens of different manufacturers and find boards that are close to (or totally) identical in most of the ones using the same SOCs. In the case of AllWinner at least, chances are the devices are designed by close partners of AllWinner - I'm assuming the same is the case for MTK and Rockchip.
The upside of this is that it's not like some random little outfit have actually designed most of the internals of most of these phones - they're largely well tested designs from well established large engineering companies. Most of them are also very simple, relying on MTK, AllWinner or Rockchip SOCs that embed most of the functionality in a single package, so the boards often just break out ports and have minor supporting circuitry which is presumably a big reason why they're so popular with this market segment.