Without having physiotherapy session tennis elbow cannot be rehabilitated fully do visit an experienced Physiotherapist before resuming sports. We have network of Physiotherapist pan India for home physiotherapy session.

You should do proper warm up and cool down while playing badminton. You should do forearm strengthening and stretching exercises with proper guidance. Wear a elbow support to prevent injury.

Ya you can start gradually and  strength exercise must.

Make sure your muscles are strong enough before starting. Stretching exercise prior to other exercises. Apply cold compression after the game to decrease any inflammation.

IMHO. The seeds for any injury are sown because of the static activities that the particular body part is exposed to and the pain will be seen while doing the dynamic activities. Because of this we stop the dynamic activities. But we continue doing the static ones like holding a mobile, working on computer, driving and other such activities. These activities on their own are more demanding.

Slowly start strengthening exercises for the extensors

A broader grip to minimize gripping effort, good shoulder rotations....internal and external rotation, lock the wrist don't be too wristy initially...don't overgrip... Corticosteroid injections initially is fine. However if continuing, collagen deposition (healing) is disrupted as recorded in recent research. Therefore not used internationally unless absolutely necessary.

You are ready to return to your former level of activity when: -Your injured elbow feels as strong as your other elbow. -Your elbow is no longer swollen. -You can flex and move the elbow without any trouble.