The former badminton player and national coach who won the All England Championship is:

Pullela Gopichand
Prakash Padukone
Saina Nehwal
P.V. Sindhu

The correct answer is: a) Pullela Gopichand.

Pullela Gopichand is a former Indian badminton player who won the All England Open Badminton Championships in 2001. He is currently the head coach of the Indian national badminton team.

Prakash Padukone is a former Indian badminton player who is considered to be one of the greatest badminton players of all time. He won the All England Open Badminton Championships in 1980 and 1981.

Saina Nehwal is an Indian badminton player who is the first Indian to win a medal at the Olympics, a World Championship, and the World Junior Championship. She has won the BWF World Tour Finals twice and the BWF Superseries Finals five times.

P.V. Sindhu is an Indian badminton player who is the first Indian to win a silver medal at the Olympics. She has also won the BWF World Tour Finals and the BWF World Junior Championship.

Pullela Gopichand is the only Indian badminton player to have won the All England Open Badminton Championships. He is also the only Indian badminton player to have won a medal at the Olympics. He is a national hero and an inspiration to many Indian badminton players.