A landfall ability triggers whenever a land you control enters for any reason. It triggers whenever you play a land, as well as whenever a spell or ability puts a land onto the battlefield under your control.

A landfall ability doesn't trigger if a permanent already on the battlefield becomes a land.

Whenever a land you control enters, each landfall ability of the permanents you control will trigger. You can put them   on the stack in any order. The last ability you put on the stack will be the first one to resolve (As a result, you can have those abilities resolve in the order of your choosing.).

Omnath's landfall ability has no effect each time beyond the third it resolves in a turn.

Omnath's landfall ability always uses the stack and players may respond to it. It isn't a mana ability because the event that causes it to trigger isn't a mana ability, even if it's the second time the ability has resolved in a turn.

In a Two-Headed Giant game, Omnath's second ability causes the opposing team to lose 8 life the third time it resolves. Each other player's planeswalkers—including your teammate's—are dealt 4 damage.