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Yes, and many Minnesota dogs have been injured or killed by lethal traps and snares. For example Scott Slocum's dog, Phillip, killed by a body-gripping trap on the frozen marsh behind their home in White Bear Lake, MN in 2012.
More information in articles like the following:
http://minnesota.cbslocal.com/2012/01/26/minnesota-man-says-dnr-hidden-traps-are-killing-pets/
http://minnesota.cbslocal.com/2012/01/28/dog-owners-speak-out-against-killer-trapping-method/
http://minnesota.cbslocal.com/2012/02/08/man-uses-cpr-to-save-dog-after-hunting-trap-accident/
https://www.cbsnews.com/minnesota/news/close-call-prompts-plea-for-body-grip-trap-redesign/
From 2012 until about 2021, the organization Dog Lovers 4 Safe Trapping MN advocated for improvements to the Minnesota State trapping regulations. The 2015 campaign was the most successful; although that year's bill wasn't heard by the MN House, it was approved by all of the required MN Senate committees. Unfortunately, it didn't advance to a Senate floor vote.
Lethal traps can be set relatively safely above the reach of dogs, or underwater where dogs don't swim, or in enclosures that prevent dogs from reaching the traps. Snares can be set relatively safely when their springs are removed, their non-relaxing locks are replaced by relaxing locks, and they're set to prevent tangling, hanging, or other complications.
Of course, in some cases, lethal traps and snares are necessary for nuisance-animal control. In these cases, professional judgement is required, to avoid sets in locations where dogs or other non-target animals might be injured or killed.
These were the principles behind the DogLovers4SafeTrappingMN bills at the Minnesota State Legislature.
No. In fact, every testimony by representatives of DogLovers4SafeTrapping began with a statement like the following: "I'm not opposed to trapping, I respect the right of trappers to be out there, doing what they love to do, just as much as I respect the right of hunters like myself to be out there, doing what we do. I only ask that they show us the same respect, by making their sets safe for our dogs."
No. In fact, the organization's president and chief legislative architect was a recreational fur trapper with expertise in (non-lethal) leg-hold trapping. In addition to guiding the main campaign of DogLovers4SafeTrappingMN to protect dogs from lethal traps and snares, he advocated for the increased use of leg-hold traps. The two goals were indirectly related: if the DogLovers4SafeTrappingMN bill had passed, some trappers, when targeting some furbearer species (for example, bobcats), would probably have switched to leg-hold traps. In other words, it wasn't an anti-trapping campaign. Its only goal was to protect dogs from lethal traps and snares.