How can I smoothly flatten select parts of a wire design?
The flat parts of the wire are achieved by using a curved face chasing hammer and a hammering block. The chasing hammer has a slightly convex surface that makes those nice smooth transitions from the flat part of the wire to the regular rounded part of the wire. If the hammer surface is too flat you'll see hammer marks on the wire. Visually check the hammer face before buying to make sure that it is convex (slightly domed) and not fully flat or worse, concave.  

Here's what one looks like:

 

The [my] hammering block is a smooth square of heavy steel about 4" x 4" square and quite heavy. The goal is to have a heavy and very smooth surface that's harder than the wire or metal being hammered so the surface doesn't get marked up opposite the side being hammered. 

Steel wire is very hard to hammer and may be as hard or harder than the hammering block. Aluminum wire might be softer but I've heard it's brittle and might not take hammering well. Most of my experience has been with sterling silver, fine silver, brass, and copper. 

The example photo is lovely but I was taught that it wasn't a good idea to hammer where wires cross as this makes weak spots in the finished piece. Maybe something has changed.