The second issue is called “shimmy”. It is caused by the front wheel oscillating around the neutral position, and there can be many causes. If it happens with all riders it could be the the front wheel is out of true but or that the fork is slightly bent and the steering angle is impacted, or tires are worn and tire slip is occurring in a cross-Wind.
It could also be some weight hanging on the handle bars like a water bottle or handle bar bag where the contents are shifting back and forth.
Also, different riders have different center of mass and probably different steering response on the handlebars, so you should find a hill and ride it down to see if you can reproduce the shimmy.