You can't really save a sponge cake that's already collapsed after baking - it's too late by then, because the damage comes from mistakes made during mixing and baking. But here's what people often miss: the temperature of your ingredients before you start whisking matters hugely. Eggs should be at room temperature, not straight from the fridge - cold eggs take longer to whisk and don't get as fluffy, plus the air bubbles end up bigger and less stable. When the batter heats up in the oven later, these bubbles burst unevenly, and the cake sinks.
The second thing that gets forgotten in a lot of old recipes is baking time and temperature. If your oven is too cold or heats unevenly, the batter takes forever to warm through and the structure doesn't set before the moisture starts evaporating. So the top looks done but the inside is still wet and unstable. Try preheating your oven about 20 minutes before you put the cake in, check the actual temperature with a thermometer (the dial often lies), and watch carefully - the brown crust will appear 5-10 minutes before it's completely done.
As for cooling - yeah, a sudden chill will make it worse, but that's just a secondary issue. If your batter was whisked properly and baked well, it'll handle a quick temperature change just fine.