Rods are always in the core. To start a reactor that is shut down (with the rods are all the way on the bottom), you withdraw them slowly until the reactor is self-sustaining. From there, you increase power by increasing steam demand (as described in the parent comment above) and continue raising rods to increase or maintain temperature.
When the reactor is operating at power, the control rods are used primarily to 1) control steady state coolant temperature and 2) provide a safe and reliable way to shut the reactor down quickly (by dropping them to the bottom of the core -- this is called a reactor scram). If you have a short-duration power transient for any reason, you can "shim" the rods in to prevent a power spike that might cause a protective action to occur (you shouldn't really ever have to do this except for during emergency drills).
If the rods were drawn outside of the fuel region at power, they wouldn't be able to absorb any neutrons and wouldn't give you any way to control temperature or power. During some specific maintenance when the reactor is shut down, you sometimes might pull one rod further out for testing.
Your question on uneven burning of fuel is insightful. That can happen, and it's caused by an uneven neutron flux (# of neutrons traveling through a unit surface area per unit time) distribution. The core designers take rod positioning into account when determining how to distribute fuel throughout the core in order to maintain a "flat" flux profile.
This is completely safe (compared to spent fuel), but how do you get the reaction started? do you have to "light" it with a neutron source when you're ready to use the fuel for the first time? or do you "light" it with radioactivity from existing fuel? or a neutron reflector?
In How-it's-made they didn't say anything like "the fuel assemblies are shipped to power plants with graphite moderators to prevent unwanted reactions during transit", so obviously there's no danger of an unwanted reaction outside of a reactor. So what kicks it off?
U-235 (the fuel used in naval reactors) does undergo spontaneous fission, but not at a rate high enough so reach criticality. Like one of the other posters mentioned, you can make it easier to achieve and maintain criticality by changing the shape of the core (so that fewer neutrons leak out) but in general you do need a neutron source inside the core that is just always spitting out enough neutrons to help the reactor achieve criticality as the control rods are withdrawn.
Once the core has operated at power for long enough, some core materials become "active" (from irradiation) and may help contribute to the neutron source.
Straight U-235 is fairly safe (iirc), but even then I don't think you'd want to ship the fuel assemblies with any moderator as moderated neutrons are what make fission more likely.
The majority of PWR fuel assemblies have similar axial-burnup shapes – relatively flat in the axial mid-section (with peak burnup from 1.1 to 1.2 times the assembly average burnup) and significantly under-burned fuel at the ends (with burnup of 50 to 60% of the assembly average). Figure 1 shows a representative PWR axial burnup distribution. As is typical, the burnup is slightly higher at the bottom of the assembly than at the top. This variation is due to a difference in the moderator density. The cooler (higher density) water at the assembly inlet results in higher reactivity (which subsequently results in higher burnup) than the warmer moderator at the assembly outlet.
Quoted from "ORNL/TM-1999/246: Review of Axial Burnup Distribution Considerations for Burnup Credit Calculations"
There are several strategies to provide additional control so it's not all at the bottom. For instance, in a PWR, reactivity decreases with temperature, so additional coolant can be injected where additional reactivity is needed. In the RBMK, a small number of rods were inserted from the bottom to provide more axial control.