Yes and no. Since the LED is more efficient, more of the power will go into light (thus less into heat). The light is eventually turned into heat, so yes, in the long run, the heat output is the same but less heat will be dissipated in the fixture.
Yes, the LED would be "60 watt equivalent", or some such. It'll also state the actual wattage but it will be significantly less. Depending on the particular LED, this can vary a lot.
The problem is that LEDs are semiconductor devices so can't tolerate much heat. Every 10C rise in heat will probably reduce its life by
2.5x (or perhaps even more).