Ok let me throw this out there....my feeling on open unsecured Wi-Fi is, if you are leaving it open and push your signal out in the open air, it's ok for me to use it. It's not like you are hacking his secured router or that you have to actually go onto his property to get the signal. That being said, the courts may or may not agree with this...there will have to be some major court rulings to set a precedent for this. Right now it isn't looking good.
That all being said, if a person is ignorant enough(ie: not informed) to not secure his wireless connection if he doesn't want other people using it than he probably isn't smart enough to capture the traffic flowing across his connection to find out who you are and see what you're doing.
If you wanted to dive into this gray area and assume it was safe and legal, yes there is probably some pretty easy ways to try and grab ahold of his signal at a much better quality than it currently is at your house now. One easy way would be to buy something like this....
NewEgg Link There are tons of different wireless adapters like that, which will help hone in on those distant signals. Heck I've even seen plans online to build things like that very cheaply.
The point is, yes it can probably be done. Whether you do it or not is up to you.
As for Verizon not being available, give them a call and make them tell you why. Explain that you talked to your neighbor down the road and he's got it. But yes you are correct that you can only be so many feet from the remote terminal because the quality starts dropping off and then becomes unusable.