What sort of machine or tool would I use to cut them all down flat to be even with each other?
First check your wall is flat, many are not. I'd use a ruler that is longer than the longest dimension of the artwork and place it edgewise (not flat) against the wall in many horizontal, vertical and diagonal orientations. Many walls are not flat.
Do the dowels need to be flat and even?
I'd consider adding three screw-in feet to the back and adjusting those so it looks right even though none of the perimeter dowels are touching the wall. If you don't like the result, you can always unscrew them and go back to plan A
One or two of the feet could be screw-in eyes or hooks for supporting the artwork. They need to be screw-in so that you can adjust how much they stick out in order to level the artwork parallel to the wall
Three feet rather than more because three feet always sit flat on uneven surfaces, four or more are likely to rock back and forth.
Machines / Power tools
As others have pointed out, large fixed power tools can be dangerous, especially when used on inhomogenous composite pieces with stepped surfaces.
The sorts of machines you might use can have the tendency to fling your artwork at very high speed into the wall or into the face of onlookers or operators. This is probably more likely when working with pieces that have uneven and stepped surfaces.
Even smaller hand-held machines can get out of control and either break bits off your artwork or off you.
An alternative might be to glue several sheets of 60-grit sandpaper to the flat top of a workbench and move the back of the piece over that in a figure of eight pattern until flat. This would be a lot of work and might cause some break-out at the edges.
Using a saw to cut a perfectly flat very thin slice from the base of thin board is something I would find very difficult to do. Particularly on a larger piece. I wouldn't attempt this.
You might be able to nibble slowly at individual dowels one at a time, each time choosing the one that sticks out furthest. Flush-cut saws might be a good choice for this.
Thoughts for next time:
The top piece in your picture looks like it is constructed on a dark rectangular backing board of some sort. The W may be too. That would avoid this problem.
If I were to make a wall-art sculpture of dowels alone, I think I'd glue it up either horizontally with the backs of the dowels resting on a plastic sheet laid on a very flat table or vertically with the dowels supported in temporary framing on the wall in the piece's intended final place.
Footnote: This is an old question but it seems to be getting attention lately and I felt it deserved an attempt at an answer. The reason it went unanswered for so long is probably that there isn't really a good answer that directly addresses the question as stated.