My name is XXXXX XXXXX I will be helping you today. Most questions will involve troubleshooting, and usually take many posts back and forth. Post back with questions as needed.
By normal are you talking about the vibration/shaking at idle?
That shakey idle is pretty typical, as it is a single cylinder engine. There is nothing wrong with the engine as long as it smooths out when you rev it up. If you were to compare it against another BF5 (and you should do so if you do not believe me) you would see they are the same.
As far as it not running on the original spark plug goes. If it was ever transported the wrong way and oil got to the plug that would cause it to fould out. Also, you can have bad spark plugs, even new out of the box, that is not all that uncommon either. But from your youtube video, it does look normal. If you youtube search videos for another 5 hp you would see they are the same as yours. If you youtube search for BF8's you would see how much smoother they are because they are twin cylinder engines, and twins idle much smoother than single cylinder engines.
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The first video the motor is above idle, so it is smoothing out like it normally would.
The 2nd one looks like it has vibration to me.
Look at this one;
Before he throttles it up you can see the vibration.
Now you can try adjusting the idle mixture screw, that might clean it up a little bit, and get rid of some of the vibration. But you will always have some vibration. That is going to be the nature of the beast with a single cylinder 4 stroke engine. At the same time, the idle mixture screws are capped over by the factory because the EPA mandates that they are. You would have to drill and pull the cap. Being a brand new engine you would probably loose your warranty. So it might be in your best interest to bring it back to the dealer and have them do it, that way it does not affect your warranty.
Also, as far as the carb jets go. That is an issue with all 4 stroke engines. The carb jets are tiny. And the smaller the engine the smaller the carb jets are.
The rings are likely fine, as you typically do not see issues with rings on the hondas.
Post back with questions