I am a new to photography, and just got a Nikon D3300. I would like to know if my camera supports high speed sync and if so, what are my options for radio sync systems? Do all speedlights cooperate with high speed hot shoes? I already have the TT560 and on the specifications it mentions flash duration from 1/300 ~ 1/20000 sec but getting it to work as the main flash on my camera's hotshoe, it fails after 1/320 sec to light the lowest part of the image.
No, the D3300 does not support high-speed sync, (none of the D3x00 or D5x00 bodies do) so you're limited to 1/200s and below shutter speeds. Anything faster, and you will have black bars on the image. In addition, your (one assumes Neewer) TT-560 is a manual-only flash and cannot perform TTL or HSS, which is why it's so super-cheap. It only has the "sync/fire" pin on the foot. For full iTTL/HSS flash on a Nikon body (or radio triggers for that matter), you need a flash that speaks the Nikon hotshoe protocol, and has all four Nikon pins on the foot.
Flash duration is not about your shutter speed. It's how long the flash burst duration is when the flash fires. The range is because the flash can be set to different power levels. But the amount of energy delivered is being controlled by the duration of the pulse. So the TT550 at full power has a 1/300s burst, and at 1/128 it's 1/20000s long.
High speed syinc in the case of your camera, is not limited by your camera, but by your flash.
So the answer is yes, your camera can do high speed sync (teoricly, any camera), but your flash don't.
A High speed sync flash makes a series of flashes to make a longer duration light. In this case the point is not to have a brief fast single output but a longer one.
I can not recomend any specific flash + combination, but you can google reviews on https://www.google.com/search?q=High+speed+sync+flash+review