Could be that Kylo picked up on Finn freaking out through the Force, given it was a very strong emotional reaction (which at least in Legends could be picked up by other Force-sensitives, the psychic equivalent of somebody yelling loudly), and something that was completely out of character for a stormtrooper. He probably sensed it and ascribed it to a "huh, that's not right" type of sensation. It's more likely out of idle curiosity (and a chance to mock Hux about the training program not being quite as perfect as the young General claims) that he looked up who that trooper was, thus learning FN-2187's designation. When Finn went renegade, Kylo made a Force-assisted conclusion about who the stormtrooper helping Poe was, possibly upon the idea of there being no such thing as mere coincidences.
As for Maz handing the lightsaber off to Finn, she knew Han and Chewie would balk at the notion of them using it, as they've got a long-established preference for a heavy blaster pistol and bowcaster respectively. Plus, Finn was probably described as being a close friend of Rey's by Han, so Maz figured that a close friend should hold onto it until the 'saber could be given to the "proper" owner. To say nothing of him not having a weapon, which Han and Chewie already did, so it'd make sense that she'd hand it off to the one guy that was unarmed.
Thus far, all the official material makes zero mention of Finn having any ability with the Force, just that he's well-trained due to his stormtrooper upbringing. In contrast, there's plenty of mentions of Rey being Force-sensitive in the various official material.
Much like Poe, I think having Finn turn out to be Force-sensitive would be something of a cop-out. I'd prefer to see him remain a "muggle" as it were, and instead just simply become a badass soldier that can dish out kill shots like they were going out of style (like he did upon picking up that First Order trooper rifle during the battle at Maz's palace).
