Can I present my Health Card and coupon together and get the additional discount? For example: 10% on presenting the Card and the extra 5% from the coupon.

No. Both the benefits cannot be combined. Medall HMP card benefits include Tests worth Rs. 11,000 under Option 1 package and Option 2 package (you have to choose one of the options) and Medall HMP coupons offer additional value of close to Rs. 24,000. Both these cannot be combined and have to be used separately.