I defend Battlefield with Glasc Mixologist vs a 6 might unit. Glasc dies then I take back a 3 might unit from my trash. Can I play the 3 might unit on the same Battlefield? Does it participate in the same showdown? And is the 6 might unit healed before or not?

To answer your questions regarding the interaction with Glasc Mixologist during a showdown:

1. Can you play the 3 might unit on the same battlefield?

Yes. When Glasc Mixologist dies, its Deathknell ability triggers. According to Rule 322.3, Deathknell abilities are added to the chain as pending items. When you resolve this ability, you are instructed to play a unit from your trash. Since you control the battlefield where the Glasc Mixologist was (and it is currently contested), you are permitted to play the unit there.

2. Does it participate in the same showdown?

Yes. Because the showdown is still in progress (the combat damage step has occurred, but the resolution step/cleanup has not yet finished), the new unit enters the battlefield while the showdown is still active. It will be present for the remainder of the resolution phase.

3. Is the 6 might unit healed before or not?

No, it is not healed yet. According to the sequence of events in the Resolution Step (Rule 444.1):

1. Combat Cleanup occurs first (which includes healing all units).
2. However, your Deathknell trigger is processed as a "Pending Item" on the chain (Rule 322.3).
3. The cleanup phase must complete as a whole before pending triggers (like Deathknell) are put on the chain and resolved.

Sequence:

• Combat Damage Step: Damage is dealt (Glasc Mixologist dies).
• Resolution Step: The game performs Combat Cleanup, which includes healing all units.
• Pending Items: After cleanup, the Deathknell trigger is placed on the chain and resolved, allowing you to play your 3 might unit.

Summary: The 6 might unit is healed during the Combat Cleanup phase, which happens before your Deathknell ability resolves and you play your new unit.