Check with your local authority on what the code is for all of the questions you asked. They'll usually have all that stuff available online.
When I did mine, it was easier than I thought. Rather than buy the super-expensive underground cable, I ran some wet-rated cable through flexible black hose (ABS or PVC? ....the black stuff you'd normally use for water lines around the farm), which satisfied the code fine at a fraction of the cost.
My service was 15 amp, so I did not need a box at the coop, I just brought it up out of the ground in some grey PVC conduit, and through a hole drilled in the coop wall. From there into a basic junction box, and split off to a simple light switch to power my thermostat and heaters, and a non-switched socket to my heated waterer.
Inside the coop, I used nothing but the metal-shielded cable to make sure nobody pecked their way to an electrifying death.
Next year, my wife wants Christmas lights on the coop, so I may have to split off another line on the outside of the coop. Yeesh.