In my experience, the granting of variances is based on need, not want. One is much more likely to obtain a variance if they had nothing to do with creating the circumstances that necessitated it. So having dogs would not be a good reason to justify a variance. Coyotes doesn't seem like a good reason either because they're not likely to come around, except infrequently and mostly during the night.. You can go inside where they won't hurt you and they're not likely to hurt the house or yard either. If 6' height solid fences in the front yard are not the norm in your area, it's highly likely that neighbors would strongly object to your having one, as it would be a major visual disruption to the "theme" that exists in the surrounding yards. The variance process requires notification to neighbors and a public hearing where they can voice their opinion about what you're wanting to do. Consider the likelihood of if they'd want to see such a fence permanently in your yard.

Almost universally, 6' privacy fences are allowed in the back and at least a portion of the side yards. Why wouldn't that work for you?