Thats soooo great that you will not have to wait until April to join EK. I bet you are very excited. It does not seem that far away now!
In regards to your confirmation, I just checking my emails and it took EK 4 days to reply back to me to confirm that they received my acceptance of the job offer. How long has it been since you sent the email? Also remember that Friday & Saturday is the weekend in DXB, so it might take a little longer.
I am sure your medicals will be fine.
So pleased for you!