Citation? How long is the lag? Years? How long does it take to fill a bathtub? If so much ice is really, actually melting in the world and has been for years; no shoreline changes have occurred anywhere in the world, then that must be some amazing lag.
The ice that's melting today is floating ice in the Arctic and Antarctic, because it's subject to warm ocean currents. Floating ice doesn't change sea levels when it melts. The ice that matters is the glaciers in Greenland and Antarctica. Glaciers weren't formed in a day, and they won't melt in a day either.
That is correct. Glaciers melt constantly, new snow forms as moisture accumulates in higher elevations and condenses, falling as snow. The snow becomes the beginning of the glacier. It's a cycle that repeats. It is repeating right now, every year.