When I see someone of my family in the morning, how should I ask them?
- Hullo, have you brushed your teeth?
- Hullo, did you brush your teeth?
And what tense would fit better with an adjunct "already"?
When I see someone of my family in the morning, how should I ask them?
And what tense would fit better with an adjunct "already"?
Both are possible, but what you are really asking is "Are your teeth clean now? (as a result of you brushing them)"
Since you are interested in the present state resulting from a past action you would usually use "Have you brushed your teeth?"
Already further clarifies that you are asking about the present state, so naturally goes with the present perfect.
(But there is no grammatical error with "Did you..")
- Hullo, have you brushed your teeth? 2. Hullo, did you brush your teeth?
Example 1 refers more to events that have happened in that same morning, and example 2 refers more to events further back.
'already' goes better with example 1.