How do Americans usually handle disputes with neighbors?

We have had four disputes with neighbors over the time my family has owned our property. All four of them wound up in court...to a degree. We live in farming country, and a neighbor had a lousy fence and 40 cows. They kept breaking down the fence and destroying our renter's corn field. We chased them out once and let him know if it happened again, he'd be responsible for damages. It didn't happen once more. It happened SEVEN times more, and the last time my dad had to put one cow down because his foot had tangled in the barbed wire and the farmer didn't come to pick him up after being told for FOUR DAYS. Dad cut off the wire, but the leg was too damaged for the cow to go home. Court held him liable for the crop damage and said he had to build a solid fence. The guy said he wasn't going to build no damned fence and we could stuff it. So, the courts said we could build our own fence, and he would have to pay for it. The guy sold his cows and moved away. We had another dispute with a neighbor who let her 8 German Shepherds run all over our property, terrorizing our kids as they walked by their house after school. The township laws stated they couldn't own more than 3 dogs unless they had 40 acres. They had 10 acres. So, the township said they had to build a fence tall enough to contain the dogs. They didn't. We went back a second time because one of the dogs nipped my youngest daughter (6). The county said they had 30 days to bring the dog count down to 3 dogs and to build a fence. My husband went nuts. "Are you serious? So, for the next 30 days, her dogs have free rein to eat my kids? Would you allow that with your own family? Mark my words, if a hair is harmed on any of my children, I will hold you personally responsible!" The county changed their tune and gave them 3 days. They moved away. A third situation involved a neighbor using our back 40 for their private parking lot of junk cars. We didn't know about it while the woods was in full leaf, but come winter, we saw it. Hubby went to their home and told them they needed to get their vehicles off our property. The guy said his tractor wasn't running, but he'd try to get them off as soon as he could. Six months later, they were still there. So, hubby hauled all four vehicles to the car parts lot and made roughly $1600 after expenses. Neighbor went ballistic because those were his project cars. Judge simply said it was considered abandoned property because it wasn't on his own land. That guy ended up in a nursing home and a new family (really nice!) bought his place. The fourth situation was one of our own fault. We have a long driveway. Our next door neighbor to the west bought a large camper trailer and had a really difficult time backing it out of his property because we have a small minimum maintenance road for our main road. Twice, he backed too far and had to get a tow truck to pull it out. Hubby saw this and offered for him to use our driveway, which connects to his in a U shape, for turning it instead of the main road. Neighbor took it and ran with it. He started using our driveway all the time and would leave huge divots and ruts. Then, he started parking his camper on our yard, just off the driveway. Then, he bought a boat and IT was parked on our yard. When he bought a tractor, he thought he was going to park it there, and hubby pulled the plug. Neighbor sued. As it turns out, we didn't have a written agreement allowing him to park there, so it was considered a verbal license to use. This can be revoked at any time. Neighbor lost and had to cover our costs. We had counter sued for road damages and had a ton of pictures to show his equipment had killed our lawn (oil leaks). We won $1400.