Homunculus Horde doesn't need to have been under your control when the first card is drawn for its ability to trigger. As long as you control it when you draw your second card in a turn, that ability will trigger.

If Homunculus Horde leaves the battlefield before its triggered ability resolves, the token will still enter as a copy of Homunculus Horde, using Homunculus Horde's copiable values from when it was last on the battlefield.

In the unusual case where Homunculus Horde becomes a copy of something else while its triggered ability is on the stack but before it resolves, the token will enter as a copy of whatever Homunculus Horde is copying. (Sometimes a horde of homunculi isn't helpful.)

The token copy will have Homunculus Horde's ability and will be able to create copies of itself.

The token doesn't copy whether Homunculus Horde is tapped or untapped, whether it has any counters on it or Auras attached to it, or any non-copy effects that have changed its power, toughness, types, color, and so on.