Yes, as long as the indicator is on, it is consuming wattage in small increments. No harm no foul on the battery or electric bill, but if it is a 3+ year old laptop, you should let the battery discharge to almost 0% for longevity.
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If the charger is plugged into both the laptop and the wall powerpoint, then yes, it will keep charging.
Yes, but the amount is very little. Most modern chargers will use about 1/2 to 1 watt per day. That will cost about .15 to .30 cents per month.
yes, i think your laptop should be plugged in when not using it because it will be charged at you wont loose your work.