Thank you all for your suggestions and advice!
Just for some clarification on some replies: We live in Southern Arizona, we get rain 2x year for the most part. We have what we call in Arizona a small wash. Without getting too technical for those not familiar with the desert, the area surrounding and in our property is known as a bajada. What that means in terms of how water flows across our land is that when we get "a lot" of rain, it sheets across the land instead of percolating down, creating several drainage and wash systems. The largest wash on our property, which is the one in question, has a channel cut of about 3.5ft at its deepest, but tapers down dramatically as it reaches the fence. When it rains, that little wash really flows and picks up all kinds of sticks and debris. We know this and regularly clear the debris from the line because we don't want that hotwire shorting out. So it bugs us that he feels the need to touch our fence.
In addition, our fence is actually backset about 18" from the property line, so technically, he has to cross onto our property to tamper with our property, its NOT a shared fence.
The Sheriff told us that they could not do anything unless they caught him, or we had video. So, up went the trail cams yesterday. If there is another issue, we will have it on tape.
I agree--I HATE barking dogs. I can hear other neighbors dogs throughout the day. And that's what i'm really concerned about since they gave him a log book, I'm walking on egg shells. I'm totally wiling to exclude the dogs from the back half 90% of the time, or as much as possible. We do have a bark collar, which is currently in use. I will think about how he views liberty as opposed to me, I don't want to be a jerk. I just want him to treat me like a reasonable person. And, yes, maybe we did get off on the wrong foot.
I agree, we should go over there and try to smooth things over. We don't plan on moving and I don't want bad blood between my neighbors and I. I do believe that your neighbors are important to long term happiness and an extra resource we may be able utilize. Convincing my husband is another matter, so I've already planted that seed in his brain, and hopefully he comes around.
Many people have referenced laws, and going to court. I'd like to make a distinction. This is code enforcement. These are not "laws" these are "rules" and there are laws governing the enforcement of those rules, but these are rules not laws, and that to me is important. The "court" we would go to is one step up from arbitration. Its not a circuit, town or county court. Its somewhere in between. So I'm having a lot of trouble recognizing the authority that this body has over us.
Being left alone, is mostly in regards to overzealous county code enforcement. This is my biggest concern. Pima County is very tyrannical. I have been personally told by a representative of Pima County code enforcement, that "the codes don't tell you what you CAN'T DO, they tell you what you CAN DO." I have seen code enforcement impose leans on people properties because the property owner cleared out packrat nests and someone complained they changed the grade of the land. Pima County forced them to hire a landscape designer to revegetate their property. Even at wholesale costs, the amount of plants those people had to plant was over $2000, not including the labor of actually planting, irrigating and the designer. I witnessed this happen to 3 different individuals. They laid down and took it instead of appealing to our local legislators. So I'm VERY concerned about developing the natural swales on our property. I want to use that large wash and divert some of the water so that it flows through the swales we develop. That could be construed as changing the grade of the land. However, it would benefit this neighbor, as he is downstream. If we divert that water for irrigation on our land, the water won't sheet across their road and erode it, the water will move through the water table underground. If we are able to speak to him face to face, I would explain this permaculture principle. I just feel irked that I have to justify my actions on MY land. Its not deleterious to anyone, its actually beneficial.
Let me make this clear: this is the desert. This is a neighborhood in the desert. There is VERY little vegetation naturally. Most people don't even have fenced property. Everyone has a well and 200 ft or more is common. But for the most part, this is residental. The majority of the people out here that are keeping livestock and gardens are doing it for personal use, IF they do that. However, most people are just living out here in double-wide trailers. Its not fancy. Theres no HOA, most people out here don't know what that is.
I really appreciate all of your replies, as I continue to mull this over. I'm really leaning toward taking the highest road possible, and make a "mea culpa", go over there and talk to them. Maybe we did get off on the wrong foot. I don't want future issues, I just want to raise my family and live my life in the way I see fit. I'll head over to the Front Porch shortly, gotta finish up dinner first

Thank you all again!