Does a rooster have a penis?
Chickens and roosters just touch to mate that they do not have the "normal" reproductive external organs. So, no, Roosters don't have a penis.
The male chicken (rooster) does not have a penis like other animals and therefore there is no penetration of the female reproductive tract when mating. Instead the female inverts her cloaca (the common junction of the reproductive and digestive systems) which comes in contact with the male's inverted cloaca and receives the sperm. The cloaca is then drawn back into the hen's body and the sperm are captured. They then begin the journey up the length of the reproductive system.
Personally, after raising chickens of all types for 35 years, my hens look like hens. However, some of my roosters also look like hens. Not only can there be delayed feathering in a rooster, but genetically there is a condition called "hen feathered rooster". In delayed feathering, the rooster pattern will occur eventually. In the genetic condition, you know you've got a "hen feathered rooster" when he starts mounting the hens and exposes the penis…
A rooster has 2 testicles, which are inside his body, but he doesn't have a penis, so, NO, he cannot get an erection. There is a small bump on the back wall of the cloaca. The sperm exit through this papilla. When the rooster mounts the hen, the cloaca meet in the "cloacal kiss" and the sperm is deposited inside the hen. It all happens very quickly without much ceremony.