Do pigeons eat honeybees?

Can I raise pigeons in the same barnyard as my beehives? Do they eats bees?

Can I raise free range pigeons without hawks and owls completely wiping them out?

birds and bees as a general rule don't go together but the squab (pigeon) is closely related to the dove and neither eat a lot of bugs.
Pigeons usually are kept in a coop, so unless your pigeon house has flowers blooming in it, bees will tend to stay away, wasps on the other hand will likely see it as a good place to build a nest.

Hawks tend to grab more pigeons than owls will and unless you have a "safe haven" for the pigeons to roost in, the hawks will see them as fair game and an easy meal when they are out foraging.
This doesn't mean the hawks will get everyone of the pigeons, but they could make a fair dent in your flock.
That is why most pigeon keepers have their birds indoors most of the time.

Around here, there are plenty of free range pigeons that reproduce faster than predators can eat them. A key seems to be to have plenty of nesting sites high up around the eves of a barn.

When I kept pigeons, it was rare to lose a bird. I only released them from their pen for a few hours in the evening. Then they'd re-enter the pen through a one-way door. That worked great, until a new family member brought a cat with her, and it entered the coop through the one-way door and killed every bird.

Thanks! I can't keep them at my house in town. My land is about ten miles away, so I only get out there once or twice a week. I wouldn't want them in a pen. I think I may try some next spring.  There are feral pigeons near my house but probably less predators here in town.

I thought about building a coup inside the peak of the barn roof. It'd basically be a elevated chicken wire pen, but they could come and goes as they please.  There aren't any mature trees within 100-200 yards. I'm not sure if that helps or hurts or doesn't matter. Our conservation department used to cut trees out of the prairies to elimate roosts for predator birds. This was to save the prairie chickens. I don't know if it helped.

The nearest trees are pines. I don't think hawks or owls would use them.

It seems like a pigeon could out fly an owl?

I have a large flock of pigeons in the gatineau mountains in Quebec.

There are hawks. I keep my seniors in cages, plus a few pairs of purebred giant runts, and late squeakers over the winter. The rest are crosses so more intelligent and are free range.

They go in two cotes every night, as they want, and a few stay outdoors even in mid winter. I bought 15 swift pigeons from a local farrier when I was starting out, that had the speed and intelligence to escape the hawks he had which were more abundant on his fields than mine are in the mountains but yes at least cooper hawks will perch in pine and spruce trees awaiting a pigeon pigeon. Hmm couldn't resist! I've lost as many as 65% one particularly cold winter, but I consider it tax, which culls the weaker birds. I also have homers, a few Modena crosses, and lots of giant runt crosses. I find most of the bigger birds prefer to fly for the cotes or hide under something when the hawks are out, but yes, protection is preferred -- I have a crossbreed livestock dog, and I am around and bark when the hawks are out. My pigeons do listen to the blue jays and wait out potential threats.Onky the younger ones are slow to react to human barking -- they know I am warning them, and understand when the dog barks so it made sense. The hawks don't like it either!

Hawks will only take one at a time. Weasles on the other hand will kill a whole flock sucking the blood out of them and leaving them for dead. They will eat through framing to tear apart a well protected coop once they've decided they are getting in so a good dog is the only solution. My dog also skinned the tail off a fox trying to take refuge under a vehicle, and I don't know if it lived, but it didn't return! I've also had a migrating hawk smart enough to go under vehicles to catch pigeons, so when the weather is bad, I only let them out late afternoon for about an hour and watch them.

People's dogs are the worst.

Yes, you need to be around, and no they don't bother flying insects. They are fabulous soil builders and some of them are just plain chummy as are the starlings!
I actually enjoy a chance to watch the hawks in action, and I have almost daily visits from turkeys as well, who poop which the dog considers a delicacy!!