but would you have problem with driving it like you would a car?
I'm not sure I understand your question.
Do you mean, can a moped, or low power scooter stay travelling in a bike lane for miles?
If so, yes.
Does this mean they have to?
Not necessarily. However, if they are slower then traffic, they would be required to keep to the far right on many roads, or at least pull over if there are more then 5 cars delayed behind them.
Then again, bicyclists are not always required to use bike lanes here in California.
would you drive that cycle through an intersection to stop at the far right side in order to make a
left turn?
In some areas, yes. It's called a hook turn, and in some places is required for all vehicles.