The cap is a pressure cap, designed to release pressure in the header tank. They very rarely go wrong. The coolant on the floor means the tank is over-pressured and so coolant is being forced out.
You must have air in the system so I would bleed the radiator. There should be a bleed screw on the back of the rad on one top corner or the other (usually right hand / left as you look side).
With a cold engine (e.g. first thing in the morning) - go unscrew the bleed screw slowly.
Don't unscrew it completely.. if it comes all the way out, it'll be impossible to screw it back in while a load of coolant is gushing out all over the place..
... just back it out slooooowly until either air or coolant comes out.
If any air comes out (hissing) then that's good... in a way.... If only coolant comes out, then you don't have air in the rad'.
If air comes out let it hiss away while you sip some tea... until coolant comes out.. then tighten the screw up and top up the coolant reservoir if the level has dropped significantly (down to "Min" say).
Drive the car around normally for the day and then do the same check again the next morning. You should have no air in the rad'. Or very little. If it's very little.... do the same bleed again and repeat it again until you get no air, or you work out that there is "always" air in the rad'.
If you have, or you always get air in the rad.. then you have a head gasket fault. It may be minor.. and you might be able to drive it around for weeks, or months... but the summer is coming and one day you'll be on a long trip up the Pyrenees at +35C ambient temperature .. the beast will pack in completely sooner or later.
Water pump problems are usually manifested by the car overheating while you're driving along normally. The car doesn't circulate enough coolant through the rad' quickly enough to cool the engine down, so it gets hot and the fan comes on, even though the water in the rad' will be cool (since the pump is not circulating the hot coolant quickly enough). But your pump is new... so not much prospect that it's faulty.
When the beast is hot.. feel the hoses then.. and the top of the radiatore. If you have hard hoses and a cool top/hot bottom on the rad', then that also points to head gasket.