If I move a Traveling Merchant and with that move I play Star-Crossed, selecting my Traveling Merchant, it goes into my hand. Do I still discard and draw?

Yes — the discard 1, then draw 1 still happens. Traveling Merchant's move trigger is already on the Chain the moment it moves, and once a triggered ability is finalized there it resolves independently of its source (Rule 383.3). Nothing in "discard 1, then draw 1" references the Merchant, its location, or its Might, so returning it with Star-Crossed doesn't blank the trigger.

Order matters in your favour, actually: Star-Crossed (Reaction) resolves first and puts the Merchant in your hand, then the move trigger resolves — so the Merchant is itself a legal card to discard to its own trigger before you draw.

The only way this plays out differently is if the ability's text checked the source at resolution (things like "here" or "my Might"), and this one doesn't.