A few years ago I had a feral pigeon we had caught. I even hauled it up from Florida to Missouri without a problem and kept him for a couple of months before letting him go. Feral pigeons can be kept.
We caught this guy because he was on death's door. He was huddled up on the ledge of a dock, soaking wet with an oozing eye. I just picked him up and carried him into the car while the guy I was traveling with stared on in utter disbelief (and a little horror).
We managed to keep him for three days inside a hotel without getting caught, then brought him to a posh golf resort hotel for a convention we were going at and let him sleep on the sink. He spent a lot of time staring at himself in the mirror and drinking filtered water.
After we got up here from Florida I kept him in a bird cage for a few weeks before we let him go. By then he had figured out how to eat food (with one eye missing he lost his depth perception and had to learn how to judge distance and whatnot again, so he had trouble eating), had fattened up and was doing more pacing than cooing.
To answering the cooing question, pigeons and doves are almost identical. They are from the same family (Columbidae) and the same order (Columbiformes). Their Latin name, Columba livia, means "a dove the color of lead." More likely than not, a lot of the doves in the Bible were actually pigeons, especially since the words were used interchangeably for a long time.
So...long answer to your question - yes, they coo. My white pigeons look like large white doves, with the same funky flying noises and cooing you hear with doves.
If you want to catch a feral pigeon use the old box-and-stick trap. Put some bread under a box and prop it up using a stick. Tie a string to the stick and hide. When the pigeon goes under the box to eat the bread pull the string, which releases the stick and drops the box.
Just remember, like any animal living out in the wild they can carry diseases. I recommend dusting them for mites immediately. These are also animals that have never been handled by people, so while they are extremely friendly, it may take a long, long time before they calm down enough to be considered a pet.