If you have a "standard" B777-300 with maximum payload, you cannot carry full fuel within Maximum Takeoff Weight limits, thus the range is limited.
Ditto with 777-300ER. Most airliners cannot carry full fuel and full payload, because they exceed MTOW in that case.
Apart from structural considerations, the only way to increase the Maximum Takeoff Weight is to add BIGGER engines.
That - or accept longer takeoff runs. But since the original plane already used much of the runway, you often do need bigger engines.
The MTOW difference between a standard B777-300 and B777-300ER is approximately 30 tonnes, thus, for the same payload you may carry an additional 30 tonnes of fuel. This provides the longer range.
No.
Some of that 30 tonnes must go for the strengthened landing gear, strengthened wing and bigger engines to carry those 30 tonnes. Some, but not all, so there is some extra fuel lifted (how much?). Some of that extra fuel is burned to deal with the extra induced drag over the original range. And in the end, there may be some fuel left over to fly the extra range.