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If your Main Phase 1 is skipped because of "Timeater", you can enter your Battle Phase (even if you have no monsters on the field) and proceed from there to Main Phase 2. If you cannot enter your Battle Phase (due to another card effect such as "Great Long Nose" or "Frozen Soul"), then you must go immediately from Standby Phase to End Phase.
This effect does not specify "and sends it to the GY". It triggers at the end of the Damage Step, even if destroyed after Damage Calculation (either by battle or by effect), as long as it destroyed a monster by battle.
If your "Timeater" and your opponent's monster destroy each other as a result of battle, the effect of "Timeater" is still activated.