The DV process should not interfere with your F1 visa. If you're still going to school and continuing your education, then they should give you a new I-20. But, they should NOT give you a new I-20 because you want to stay longer in the U.S so you can wait for the interview time. In other words, a new I-20 is only issued if you couldn't finish your study based on the time designated on your I-20 or you want to continue school in the U.S and pursue a higher degree or transfer to another school.
If you're doing AoS in the U.S, then you have to keep your F1 status until the interview. But, if you're doing CP, you don't need to keep your F1 status.
If your F1 status is lost/expired, then you need to go back to your home country and wait there until you get called for interview at the U.S Embassy/Consulate in your country. If you're going to do this, make sure you tell KCC that you are doing CP.
If your F1 status is still good and it's possible to extend the I-20 because of legit reasons, you can either stay in the U.S and wait until you get called for an interview at the U.S Embassy/Consulate in your home country or do an AoS in the U.S.