Earlier this month I adopted a young (1 year old?) female dog who had been found wandering around the woods severely emaciated. I don't have any reason to believe she's been properly socialized or house trained. When I first brought her home, she was very timid and sad...but with lots of love, she's really comes out of her shell!

...Unfortunately all sorts of behavior problems are surfacing. Housebreaking is not going well, and I believe she's having all sorts of anxiety issues (she'll go 20 times all over the place as if she's pissed at me for leaving her alone.) She's extremely destructive. If there is a way to destroy something, Kazak will find it. She has tons of toys (chew toys, a kong, etc.) but she'd much rather eat the sofa. I feel like she's bored, so I try to exercise with her. When I walk her, she'll try to pull ahead, but if I actually take her on a run, she refuses.

She doesn't get along with the other dogs (we share a big fenced in backyard) so I can't leave her out there alone. After we "play" (she runs laps around the yard) I can never get her to come back in. She'll either run away from me, or lay down and refuse to budge (she's a biggish dog, husky sized.)

She's a sweet heart at her core, and I am more than willing to put in the effort needed to get these issues under control. I do work eight hours a day, but my sister lets her out twice a day while I'm at work and I take her hiking on the weekends. As for breed...not really sure. People have suggested border collie, boxer, pitbull, mastiff, lab, and greyhound. Any help would greatly be appreciated. I'm a patient person...but the rest of the household is at their wit's end.

has she been to the vet & had all her shots, a physical AND checked for a UTI?

do not let her out in the yard unless on a leash, that way you can get her back in. sounds like she is bord silly which is causing some of the destructive behavior. she needs at least two good jogs a day to channel off some of that energy on a good solid 6' leash, not one of those flexi things.
keep her close to your left leg.

have you considered crate training for her? it would be especially helpful when you are not there. she has to much freedom of space in your house.

I would also get her into some reputable obedience classes asap. they usually give the dog some self confidence & helps you bond.

doggie day care? a dog walker? google NILIF & start using it right away.

make sure the "rest of the household is in sync" there has to be complete consistency in how any dog is trained

there's a start

I would love to go for jogs with her. She refuses to run on a leash, though. She actually sits or lays down as soon as I start jogging. I would take her to dog parks, but she does not get along with dogs smaller than her.

I'm also a firm believer in crating, for the first two years in fact. It's especially helpful while working on housebreaking a dog and avoids destructive behavior while unsupervised. Basically, you'd put the dog in the crate anytime you're away or when you're totally preoccupied and can't keep an eye on her. You don't want her spending too much time in the crate though so you may need to adjust your work schedule when it comes to chores around the house so she can spend a good amount of the time out with supervision. You could also put her outside in the yard on a lead when your unable to supervise her. So long as she's tethered she can't go after any other dogs in the yard and they'll most likely just avoid her until she becomes more socialized. The important thing is whenever you let her out of the crate you have to bring her outside immediately (on a leash preferably) and wait for her to pee then give her lots of praise and a treat. Since she's gone so many times in the house already, it's important to make sure you neutralize the urine with a product like Nature's Miracle. Regular house cleaners aren't enough because a dog can still smell the urine after cleaning with them, making Kazak a lot more likely to go in the house again. If at any time she still pees in the house while under your supervision, ideally you would stop her in the act with a NO! command pick her up immediately then carry her outside to finish peeing and then praise her and reward her as usual. At one year old though she's a bit big for the carrying part so you just want to stop her in her tracks as best you can if she starts to pee and then quickly bring her outside to finish. With constant reinforcement (positive and negative) she will get the idea (outside=good, inside=bad) eventually.

As for teaching her to come to you, if she sometimes comes when you call her simply give her a treat every time she does it. Nothing motivates a dog like food so once she associates coming to you at command with getting a treat she'll do it more readily. If she NEVER comes to you, or to hasten the process for quicker results, put her on a long rope (or you can simply use the outdoor tether) then let her roam away from you (if she simply lays down then you can walk away from her a short distance). Call her by the command "Kazak come!" then pull her to you with the rope and give her a treat. Repeat the process over and over with increased distance each time. Again, once she associates coming to you at command with getting a treat she should start coming when you call her without you needing to pull on the rope.

I agree with the others, go with crate training. I used to think it was unnecessary....until I got my dog in September :roll:
She has seperation anxiety so she cannot be left alone in the house..or there will be no house when we get back! We bought her a nice crate and now she just loves it. She naps in there and usually sleeps there all night (with the door open). It immediatly relaxes her. When we leave her at home in it, she'll often refuse to come out when we get back! Needless to say I've totally changed my opinion on crates after seeing the good they can do.