If the neighbor behind you is game and wants to have a similar looking run all the way across his yard then it'd be nice to coordinate it all at one time. Same thing for the neighbor next to you.
But don't rule out having a gate there. Several neighbors have them and it's really convenient for the kids. They don't get used "all the time", as that would border on getting kind of rude. It's mostly just when the stray frisbee, ball or other toy goes over.
After doing some surveying, is really very rare when everything at "square" or even have a couple of exact 90 degree corners. - This is especially true in areas where the is some sort of magnetism where the original surveying was often done ears ago using magnetic directions or even the old "meets and bounds" system that had to modernized when the old trees, streams and rock had disappeared.
Before doing anything remotely related to property lines, make sure you know where the pins are and they have not been disturbed since being set. - It gets very costly, complicated and inconvenient to correct or adjust. After that decide if you want the fence to be on the property line and co-owned or on one side or the other where the ownership, appearance and maintenance is well defined.
I really cant see how building on the property line can be a good thing in the long run. What happens when you or one of your neighbors decides that the fence you chose looks cheap and decides to upgrade to a better product. Now that corner post becomes an issue of contention between neighbors. If it was on my property, I would not consent to having a fence installed on the property line, if only to prevent future problems.