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HAZEL - is a PERFECT name! She is adorable! 
1st - she absolutely loves peeing in a bush. I mean she will crawl right up into it and then relieve herself. This doesn't happen every time, only about 85% of the time. Is this normal for this breed?
Not uncommon. Have heard of it with other breeds too.
2nd - she will squat-walk while she is going #2. She will never just squat in one place. Is this normal?
Normal. Not all do, but poop walking is common
3rd - if it is raining out, she will hold everything in until she can't. We do try to take her out in just a mist, and it is a struggle. Only when she can't hold it anymore will she go out, and it is only to pee. She refuses to walk to her pooping ground. Living in the Pac NW, rain can be continuous for 1 - 2 days. This concerns us. Is this normal?
It isn't good, but common. My dog would hold until she had anal sac ruptures. I put her on fiber 3x a week, stool softener, and prozac. (the last was for other reasons, but once on it, she no longer had a stroke when she had to poop in the rain! ) I have even done matchsticking to make her potty. Giving fiber and stool soften simply makes her have to go, rain or shine. Having dealt with anal gland rupture, I never want to go through it again. So yes, be concerned.
4th - we have her on a gluten and grain free diet, would this be the reason why she doesn't have to poop as often? I have noticed that if we feed her a bit of dinner scraps (not a lot by any means), she also seems to have a bit of a looser pooh and will pooh 3 - 4 times in one walk.
Here's a quote
DOES MY DOG NEED A GLUTEN FREE DIET?
Unlike in people, celiac disease is uncommon in dogs. As a result, most dogs do not require a gluten free diet. The exception to this is the Irish Setter. A small number of Irish Setters have been documented to suffer from a congenital disease that results in an intolerance to gluten. This has only been reported in certain Irish Setters and only in the Irish Setter in the U.K. These dogs, however, will benefit from a gluten free diet.<<
The reality is the gluten free human diet has one great benefit, the few who actually have celiac disease and the VERY tiny number who don't but really need gluten free are benefiting by having self-diagnosing people think they need it. That has resulted in a major price drop and major increase in food options for those who need it. Since it won't HURT those who wrongly are sure they need it, that's a win/win. So dog food companies making gluten free are simply making a buck off fake science.
Dogs need a certain amount of fiber to form stools. Really super cheap food has tons of indigestible stuff, giving huge poop. Good quality diets have less junk, but still enough fiber to help form stools. I encourage you to give a few bites of whatever you are eating on a regular basis. Your dog will enjoy it, will have more variety, and it's good. (Obviously not chocolate, onions or things they shouldn't have.... read someone shared a xylitol candy with their dog.. that's a huge toxic no)
So grain free. I feed grain feed, even though I have always know there is no research backing it. While dogs may be allergic to a specific grain .. it's not common.
In fact: >>The top five allergic provoking ingredients in dogs, in descending order, are beef, dairy, wheat, chicken and egg.<< Corn, which I avoid as I know when my dogs have eaten corn it comes out just like it went it, ,, about 7% of dogs have an allergic reaction. Food companies poo poo that and defend it as a good choice for protein. No thanks.
However, I just read a new dna study and will be rethinking some things. They have found that dogs that developed with agricultural communities actually have a higher ability to process starches than dogs that developed (like my Samoyed) far from agrarian societies.So those dogs may both use and need more carbs than other breeds.
See if she likes any fruits or veggies and offer them with her food or as treats. Low or no sodium green beans are good also. IOW, you need to get some more bulk in her diet so she has to poop. If you can, maybe create a covered area where she can potty without rain hitting her. Before prozac, I was considering putting a cover on our walk around that leads to the back yard fencing. We were even happy if she pooped on the back porch as long as she went. So try umbrellas and rain boots.
At least you aren't alone.
