To answer your main question of whether king pigeons can fly or not, it really depends on the type and size of king. The quality show kings with shorter wings and tail and rounder body are said to be very clumsy in flight of any large distance. Many of the utility kings are also too heavy to fly any large distance, but some of the kings which are not show or utility quality and are simply kings by name and not for weight or show purposes are said to be able to fly greater distances. Overall, kings range from being unable to fly more than 15 feet in the air to being able to keep up with a team of homing pigeons for 10 minutes before becoming exhausted and landing. It just depends on the type/weight/strength and stamina of the pigeon. The male king pigeon cross in my loft (probably 25% homing pigeon blood, he was a rescue) can fly to the rooftop with relative ease and keep up with my racing team for about two minutes or so before becoming exhausted and landing. He is a very heavy pigeon, but has the muscle to get in the air for a little bit. I have never heard of a king pigeon which is too heavy to fly short distances in a loft, but they are slower when reacting to danger or lifting off the ground.
Pigeons can live with chickens, but as Fowlsessed and TheifPouter06 mentioned your chickens could be aggressive to the pigeons especially when first introduced and possibly spread sickness or disease. Although many people have had bad experiences with trying to keep chickens and pigeons together, my first pigeons were housed in a chicken coop and I have kept a group of pigeons with my current chickens for a few years now with hardly any problem. Maybe I am lucky, but it has worked out fine so far with little aggression after the chickens adjusted to the little colorful birdies walking and flying around and I have seen absolutely no signs of sickness from the pigeons housed in my chicken coop. If kept with chickens, pigeons will need a separate feeder and water dish/fount that the chickens cannot get into or knock over, adequate space both on the floor and on perches, and either do not allow them to nest in the chicken coop or they must have a completely secure and chicken-proof nesting compartment. If any chickens get to the nesting area especially when there are eggs or young, the chickens do not completely understand what they are seeing and will wreak havoc.
Again, although I have heard many other people say they have had problems feeding their pigeons chicken scratch feed, I have fed my pigeons scratch (and layer pellets) on occasion before with no problem whatsoever. If feeding that as their main diet, they would probably enjoy some other seeds or grains mixed in from time to time such as millet, a few raw spanish peanuts or some field peas to give them a variety. Just remember not to feed pigeons scratch or layer layer feed (or any other feed, for that matter) which has been exposed to too much moisture or which has gotten wet, as it can spoil quickly and easily make pigeons very sick.
King pigeons, if tamed and handled, can become a very friendly pigeon towards humans. Generally they are docile towards other pigeons, but there are always those territorial birds who can't stand other pigeons in their space and are more aggressive. It partly depends on the personality of the individual pigeon.
Hope this helps!