Paid a faceless man with what? They don't typically accept gold. We did not necessarily see that with Arya, she was freaking out and probably hallucinating.
There's the scene with that blonde girl when Arya is spying on Littlefinger. LF hands her what looks to be a Braavosi coin like the one Jaqen gave Arya. She then says to him "your time is up." She also has blue eyes like Jaqen if that means anything.
LF also said earlier in the series that his family was originally from Braavos (great-grandfather), so he could very well have connections there.
It also assumes Littlefinger is basically flawless and sees everything that's coming.
Not necessarily. Bran repeats chaos is a ladder to LF. Dude freaks out knowing that Bran has super powers. Figures out a way to fake his death and get out of WF. Seems plausible to me.
It certainly would be an inventive story, but I'm not so sure that the writers would put in such a twist this late unless LF has a huge role in the end game. Also, if anything I've learned that sometimes fan theories are more interesting than what actually ends up happening.