I had a friend who was practicing with his new .416 for his upcoming Cape Buffalo hunt. He was using it to shoot out of his second story window at ground squirrels in his pasture. Any squirrels that he missed got new holes to hide in.
To specifically answer your question, it is my understanding that the .22-250 was originally developed as a varmint cartridge, and with the right bullets it has been successfully used on deer sized (and larger) animals. I've shot both mule deer and antelope with mine. I have friends who have shot elk and buffalo with theirs.
And the .25-05 was originally developed as a deer cartridge that would also work well on long range shots on varmints on windy days. I've used my .257 AI, which is balistically similar to the .25-06, on everything from prairie dogs to elk. It's my favorite deer, antelope, and bighorn sheep rifle.
When placing a trail camera don't just look for a well used trail. What you want to do is look for a freshly used trail off by itself that goes from a north facing ridge, thick forest, brushy knob or some other similar bedding area to a food source. Don't forget water sources. Especially in the summer months the deer need water so look for a good trail going down too a creek surrounded by thick cover and place the camera 100 yards up from the water source.
Scent control is very important...