No black spots/splotches = not a Blue Slate.
The turkey in the middle does look like it has some white feathers. It may be due to the poor quality of the photo. If it has white feathers it is a Mottled Slate. If it does not have any white feathers and it does not have any black markings, it...
It is possible that there isn't any white on it but I can't tell from the current pics. Your description of it's poult color sounds like a Self Blue aka Lavender.
Blue Slates don't breed true. If you have a Lavender and breed him to a Blue Slate hen, you will get 50% Blue Slate and 50% Self...
Show a full photo. The images you have shown look like there are some white feathers which Blue Slates do not have but Mottled turkeys do have.
I have raised top quality Blue Slates for many years and am aware that not all "hatcheries" are ethical. I know of hatcheries that sell Self Blues...
Yours appears to be a sinus infection which needs treated.
I can't see enough of your turkey to be sure, but it is not a Blue Slate. I can't tell from the pics whether it is a Tri-color Mottled Slate or a Mottled Slate.
Show pictures. It is likely caused by fighting. It is not really common but on occasion the fighting causes an air sac leak. This causes the affected area to swell up like a balloon. The cure is to lance the site and let the air escape.
We need to see pictures so we know whether or not this...