If Finn is not related to any Jedi, then why did he fight with the laser sword and why did Kylo Ren feel him when the First Order were invading the village at the beginning of Star Wars Episode 7?
Force abilities and sensitivities are not, Skywalker family aside, inherited. You don't have to have force-capable parents or grandparents or ancestors to be able to use the force. As the Prequel Trilogy shows, force abilities are just a subset of having a high count of midichlorians. Jedi's normally went around, looking for kids that have this tendency, to train them. Their parents typically don't have Force capabilities. Anakin's mother did not either. In The Force Awakens, we learn that Luke had a bunch of (now presumed dead) students to train in the Force. Students who likely did not have any connection to Luke or any Jedi, as the Jedi all died decades before the movie takes place.
We also don't know who Finn's parents are, so we can't even say 100% for certain that his parents were not Force users.
All we know is that Finn is Force sensitive. The full extent of his Force powers are not known. As the movie shows, he did not do anything close to the feats of Rey, Kylo, Luke, Leia, or any other known force user.
As to how he could use the Lightsaber, there is conflicting info on if you have to be a trained Force-User to be able to. Some aspects of it do require the Force, like deflecting blaster shots blindfolded (As Ben taught Luke to do). Some say that you have to use the Force to proficiently use a Lightsaber in battle. Finn fought a Stormtrooper, and a badly injured Kylo Ren. Not really a duel between expert lightsaber duelists.
We don't know. He was, afterall, a stormtrooper. The reason he might have Force sensitivity is that in the movie, it is revealed that stormtroopers are not clones anymore, but are stolen as children. So Finn was taken from his parents at birth, and we don't know who they are, so in that respect there's just not enough information to say for sure.
Yes, he does use a lightsaber to some level of proficiency, but honestly, he doesn't use it very well. In the movie, he would have died twice had there not been outside help. In fact, he does almost die anyway. He got through his fights with nothing but dumb luck and outside help. Also, a lightsaber is just a sword, albeit a very nice, powerful, sword, but a sword nonetheless. It does not take any extra skill to wield a lightsaber than a normal sword. The skill is in creating a lightsaber, which is a Jedi Knight's last test.
All in all, we just don't know enough about Finn's backstory to say yet, although I'm sure that in the episode VIII there will be a reveal if he is at all sensitive to The Force.