I was shocked when I saw it attached to the trailer like this. Can this cause damage to the suspension....or anything else?

 

Am I missing something? Looks like the right way to attach it to me. The tires/wheels are locked to the trailer deck. If you were pulling on the frame, that could compress the suspension and could be a problem.

 

When my cars have been flatbedded the tow tech has used a webbed setup that fits around and over the top of the tire, then is lashed to the front/back/side of the towbed platform. With the E any webbing has to be used carefully, esp on rear axle due to brake lines running along those areas.

On this car, (red convertible) it looks like the webbing is attached at the outside, run through the wheel and then to the back to be fastened there.. I would assume that the right hand side was done in similar fashion.
In this photo, it shows the straps 'around' the tire, but probably because there are no spokes in the wheel.