Muskrats can absolutely remain active under the ice. I think of them like mice....if you see one, well, you get the idea. If the runs are active you should notice a lighter area of silt on the bottom. The run itself may or may not be muddy, but the light colored area against the typically darker pond bottom is a pretty good indicator.
Conibear 110's are the usually employed traps for rats. The dog is a legitimate concern, especially if he likes to get in the water and dig. Removing the traps every morning is doable I suppose, but it would be a huge PIA, and might spook the rats' into digging another burrow.
If you have a safe place to shoot, I would take a scoped .22 and find me a comfy spot that overlooked a good portion of the pond. Grab a beverage of choice and head down there about 30-45 minutes before dark. They tend to get active about that time. After a few nights keeping vigil, you should know if you have a rodent problem.
BTW, dogs usually LOVE to eat muskrat. I don't know what it is, but they will readily consume them.
