Does a unit have to be holding Dusk Rose Lab for its ability to trigger? If it isn't conquered will its ability still go to the stack during the beginning phase?

No, a unit does not have to be "holding" Dusk Rose Lab for its ability to trigger, nor does the battlefield need to be conquered.

According to the card text, Dusk Rose Lab is a Battlefield card with the following ability: "At the start of your Beginning Phase, you may kill a unit you control here to draw 1."

Here is the breakdown of how this works:

1. Trigger Condition: The ability triggers automatically at the start of your Beginning Phase, regardless of whether you have conquered the battlefield or are currently "holding" it. The only requirement is that the battlefield is under your control (as it is a battlefield you are using).
2. The "Chain" (Stack): When the Beginning Phase starts, this triggered ability is placed on the chain. Because it says "you may," it is an optional triggered ability. You choose whether to place it on the chain when the condition is met.
3. Requirement: The only requirement for the ability to function is that you must have a unit you control at that battlefield location to kill as a cost when the ability resolves.

As confirmed by RiftJudge FAQ #9428 and #9473, this ability triggers at the start of your Beginning Phase, and you have the opportunity to resolve it then. You do not need to have conquered the battlefield to trigger this effect.