- Are there any implementations of nanopayments that already exist?
I don't know of any however unless you used that project's source code it wouldn't matter.
Does the fact that I didn't come up with the nanopayment protocol restrict what licenses I can use?
Unless you used someone else's code you can license it however you want. If your code is derived from another project's code then you are bound by the copyleft provisions (if any).
If I start with GPL v3, I can always relax the license to be more lenient, right?
No. You can however dual license the project (simultaneous commercial license and GPL license).
GPL is a license with a strong copy left provision. It requires that all derived works also assume that license.
GPL would be pretty toothless if you or someone else could simply derive from it (release an updated version) and remove the copyleft provision.
Any other implications of choosing a license that I should consider?
If you don't understand the implications I would recommend doing some research.
You can always use a more permissive license like MIT which would keep the project open source but allow just about any type of derived projects (even closed source proprietary ones).