Yes, a copy retains all abilities of the original card except any modifications specifically noted on the source that created the copy. In the case of the Helm, the copy is not Legendary if the original is. When the creature token "enters the battlefield" it's ability will trigger.
706.5. An object that enters the battlefield “as a copy” or “that’s a copy” of another object becomes a copy as it enters the battlefield. It doesn’t enter the battlefield, and then become a copy of that permanent. If the text that’s being copied includes any abilities that replace the enters-the-battlefield event (such as “enters the battlefield with” or “as [this] enters the battlefield” abilities), those abilities will take effect. Also, any enters-the-battlefield triggered abilities of the copy will have a chance to trigger.
The way your question is worded, the answer is "no". When the copy of Rakdos, the Showstopper enters the battlefield, that copy would have a coin flip ETB, and that ETB would trigger. The ETB of the original Rakdos, the Showstopper would not trigger; there would be a new object with a new ETB. In this particular case, it amounts to the same thing, but if there were wording along the lines of "do [blah] for every time this has triggered", this would be considered to have triggered only once.