Do they breath well (cool in the summer)? Do they go under a motorcycle shifter without any trouble?

Can't speak to the motorcycle shifter since I don't ride, but to me they're very comfortable in the summer. I find them to be very, very close to as comfortable (both in terms of breathable and wearable) as most any shoes except running shoes. As long as I'm not being super active, I'm good wearing them any time of year.

Does this boot have a deep toe box?

By measurement my foot is at the widest point of what's considered a medium width foot and I got the extra wide and they're perfect. A medium width would have probably been to tight. But these are plenty wide and some of the most comfortable boots I own and that includes some of danners $275 boots. First day of wear I spent 14 hours in these boots outside in mud and water and had no leaks,no sore feet or anything by the time I took them off that night. I even rinsed them off with the water hose and still no leaks. I can't even tell they are composition toe. I'm female and I have neuropathy in my feet and legs so I'm very picky about shoes,boots. They have to be comfortable.