Yonex shoes are narrow, but they do have a wide shoe. Their wide shoe is a size D which is considered Medium in North America
Yonex currently has two wide shoes, the SHB 65W
and the SHB 65 Alpha Wide
The difference between the two is the arch support. The SHB 65W has titanium fibres used to reinforce the arch.
I can confirm, it will not leave a mark on the mat. The soles are made of a different material. Cross trainers or outdoor shoes use a different type of sole that would leave marks.
There is no gel on the outer soles. The gel sits on top of the sole to provide cushioning.