Can a small cut on the finger, even a paper cut , that doesn't even bleed cause the finger to have a knot or to be a little bit swollen?

Yes: The normal healing process involves an inflammatory phase which most often manifests as redness and slight swelling in the first few days. For skin cuts, burns, abrasions, etc. , if you are keeping the injured area clean, the redness and swelling should reduce significantly or resolve within a few days. If it does not reduce or is obviously worsened, there may be infection requiring medical help.