Canine Behavior/Can't leave dog alone for a minute
I have a 2y/o Dobe/shep mix that is the most gentle and lovable dog. She is also great with the kids and guests. When we are around her she is wonderful. As soon as my wife or I leave the room, she gets into mischief ie. chewing things, going on counters, getting into garbage, going into rooms that she knows she is not supposed to be in, etc. This happens even though my kids are still in the room. She even gets on the kitchen table when they are eating if we walk away. I know about correcting behavior when we catch her doing bad things but she already knows what behavior is acceptable as she patiently waits until we leave the room. As soon as we start walking back into the area, she runs away with her head down, tail between her legs and pees if you approach her. (Which makes me more upset) We use a crate when we go out without problems, for the most part. I even used the kids video monitor to try and catch her in the act with a shock collar on her but the mastermind of a deviant did nothing and outsmarted me. I don't want to crate her every time we leave the room or keep the collar on her all the time. Any suggestions would be greatly appreciated and thank you for your time.
THROW OUT THE SHOCK COLLAR. Better yet, wear it yourself and see what it feels like. The dog, unlike you as a Human, is incapable of experiencing the pain with its MOTIVE, it will merely associate the pain with what is presently in view: could be one of your kids.
From the photo, I'm seeing a very submissive and very anxious dog (ear set, tucked in body posture). Now, this could be a reaction to the camera (lots of dogs don't understand the camera and it freaks them out) but it could also be an indication of her stress levels. She doesn't "KNOW" she's not supposed to be in any room, this is far more than the cognition of a dog allows. You are anthropomorphizing (ascribing Human cognition and reasoning to another species). "Doing bad things" and "catching her" are also terms she does not understand nor can she. To her, the garbage is fun and self rewarding. The only "bad" thing is your anger or whatever other emotion you offer (body posture, words, actions, etc.) She does not connect her foraging (which is, after all, what the dog is designed to do) with your reaction. If she runs with her "head down, tail between her legs and pees if you approach her...which makes me more upset", she is reacting to YOU, your anger. This "peeing" is submissive urination and, in the dog culture, it's an instant signal to the other party (dog) of non-threat and submission. Humans aren't dogs, dogs are habituated to us as conspecifics but they have no way of understanding that these signals (like submissive urination) are not understood by us and this further confuses the dog.
The dog did not "outsmart" you as a "Mastermind" because she was wearing the shock collar. It's quite possible you had already inadvertently shocked her or the weight and unusual restraint of this collar (dogs perceive their collars as rather "important") confused and inhibited her.
Put the dog on long training leash; attach one end to you. Spend the next two weeks forcing (without actual physical manipulation) her to follow you. While doing this, "baby proof" your household. Find a trash container with a step-on device that is impossible to open unless the device is STEPPED on (and the dog will have no way of doing this). These are usually metal. The dog sees your children as equal to her in social hierarchy or, possibly (and this is normal) lower ranking. Their food is her food. Jumping up on the table is "allowable" when you are not there, since you are the higher ranking and, in the dog culture, the food belongs to the higher ranking but is then available to the next in rank (her) when you are absent. She does NOT know this is "bad". You also have NO knowledge of how your children may have participated in this behavior (quite innocently, of course, being children) by feeding her from the table or even showing hilarity (laughing) when she jumped up the first time. That would have rewarded and encouraged her.
This is a pleasant, sweet dog whose life is important, as all life is important. She is a member of your family and, as such, deserves a little special treatment right now. I think all of this is quite correctable but I can't come to your house, interview your children (without you in the room), asses the dog, observe her in the family situation, and ask questions. A little positive reinforcement training and some time redirecting the normal drives she is experiencing CAN and WILL correct these problem behaviors but without direct evaluation of the children involved, as well as an assessment of the adult attitudes, it's impossible to prescribe a treatment that will work and make the dog feel less anxious and her behavior more predictable.
I strongly suggest you find a Certified Applied Animal Behaviorist (CAAB) who CAN be there in person and assist. I doubt very much it will take more than one session and you will get a very clear understanding of the psychology of dogs in general and this one in particular. You may be able to find one from the following sites:
While the dog is on training leash, DO NOT SHOW any anger or frustration and do not "discipline" her in any way. Instead, routinely ask for an easy behavior ("sit") you can praise and even food reward (with tiny tidbit). Assessing the ability of children to establish a higher social rank with a dog means being able to see and talk to them (which I can't do) but an in-person CAAB can, and that will have remedial results (over time). Meanwhile, you can occasionally "set the dog up" by putting a fake "meal" on the table, instructing your children (the oldest must lead the youngest) to "freak out" when/if she jumps onto the table (after you ceremoniously and deliberately "leave" the room) and run out of the room screaming. If you do this a few times (staggered over the course of three to four days), the dog should form a conditioned avoidance response: connect the shock and disappearance of the children with her jumping onto the table to steal the "meal".
For counter surfing: children must be strictly OUT OF SIGHT (as I said, you don't want the dog to connect her fright response to THEM): get three to four lightweight cans, empty them, put three pennies in each, crimp in middle so pennies can't escape. Now buy strong (Bounty) paper towels. Use four to five sheets connected (so that two sheets can cover the cans) and make a peanut butter sandwich (half of one is ok) with the first paper towel PART of the sandwich (it will stick to the peanut butter), then leave the room. Dog will counter surf, she will grab sandwich, as sandwich comes down SO WILL THE CANS, making a lot of NOISE. This dog is so soft in temperament that this will frighten her. You should not need to do this more than three times (not too close together) to get an avoidance behavior from her (no more counter surfing).
What you MUST avoid is making a connection between her foraging, or "stealing" objects (such as food, napkins, etc.) with your ANGER and any discipline. The dog will become extremely fearful and fear is the absolute cause of most active aggression. It isn't fair, she doesn't have the ability to "understand" her behaviors are not acceptable. She's being a DOG. And with kids in the house, there's often a lot of hilarity and confusion and dogs are part of that mix. Some dogs have the equilibrium (temperament plus socialization and good training) to maintain a sense of security under such circumstances while other dogs do not. It's not the dog's fault. You can fix this.