Yes, you can leave it plugged in. Once in a while however, you have to use it under it's own power, at least till it reaches 80% to keep your battery in optimal condition.
As for the power cord, you answered that one yourself " for convenience "
How long does it take long to charge all depends on how low the battery was when you charged it.
Yes, it's ok to leave your Macbook plugged in all night. Just make sure you use it under it's own power once in a while. You don't have to fully discharge it which might do more harm than good.