These are very difficult questions to answer precisely, because there are no firm guidelines. Conventions change as you move between groups with common social class, education, work, age, gender, status within the group, etc.
Depnding on who I am addressing, I might use any of the following, and I might, in different contexts of situation, use any of the following to the same person. I have arranged them in a very rough personal ascending order of formality:
Yes? .....Any luck?.....Find it?
Find it? ......Found it? HaveHave you found it? ......Did you find it?
I hope you (have) found it. ...... I do hope you (have) found it .
Sometimes the choice depends on whether they have, or have not, found something that belongs to them or to me.
I don't suppose you (have) found it? does not feature on this scale of formality. That strikes me as what I might say to someone who might have lost something that was important to me. I am expressing a rather forlorn hope that it has been found.