Should I form and pour the concrete all at once?
If you don't care about leaks (or want drainage, so the dog is drier) forming the rim, pouring the rim, and then pouring the central slab is similar to whats typically done in a house basement. The joint between the rim and the slab will generally leak. If you pour a slab and then pour the raised edges on top, expect failure - if to scale, those are very thin sections of concrete and would be quite weak. You could key them with a form board into the slab face to improve the mechanical bond and thickness of the additional pour, but on this scale that's not much different from just pouring the rim first.

The usual reason for attempting to pour a structure like that all at once is to have it be waterproof - a monolithic pour will be much better at that - but's more complicated to do, as you either need to form the top of the depressed area, or (more commonly) form the rim so that the inside form is open, pour the rim (letting it ooze to the inside) and judge the slump and initial set such that you can get the top of the rim struck off without overfilling the central area, yet still be able to strike off the central area. In this case, that would seem to make things wetter for the dog, so I don't see it as an avantage.

For the easy approach, consider using 4" concrete bond blocks for the rim - no separate form to build, just lay the blocks, pour them full, and then pour the middle. Toss in couple sticks of rebar to make it robust for not much money.