Obviously these situations are all hypothetical. Yes, I plan to introduce myself and even invite the neighbors to shoot on the land as well. The neighbor south is really good people and probably the type of people you want as neighbor. The one north is the one with the horses. I've heard he's good people too but just doesn't care about fencing up when he's been confronted about it before.
Everything I've read about Texas being open range is in regards to not having to fence up towards the street, highway, county road, etc. Meaning it's okay for the animals to go into the road. If traffic runs over it, the driver is responsible. I haven't seen much about open range in regards to neighbors. I have to keep digging some more.
No, I will not shoot the horses lol. Yes if I'm ever shooting, I will be situationally aware. I know I am responsible for every single one of my bullets that leaves my muzzle.
By the way, anybody know where they have T-posts and barbed wire rolls for cheap lol.
Keep the advice coming.