The regulator is definitely not going to be enough to protect to all mess a car can make on its power rails.
You probably want a transient suppression diode, to short away any very unwanted peaks. Breakdown not too close to 12V, not too far above 30V I would guesstimate. (The double linking about the regulator confuses me a little, but you are going to want a regulator with at least an upper limit of 36V as you drew, preferably more if possible)
Some form of inrush limiting might be an idea, to share some burden with the TVS in case of extended overvoltage, calculations would need to be based on the dynamics of your loads as well.
An extra filter isn't a bad idea all the way at the begin, closely followed by an "oopsie" filter, I generally call fuse.
My late-evening lazy me couldn't quickly find a full datasheet of the pump, but I'm not looking very well right now. If the pump is a normal type reasonably rugged DC motor, as the page suggests, it should not be bothered too much by the average car-noise, provided it does handle a continuous 16V ~17V. Doesn't happen often or very likely, but it might and it could melt a very tightly specified 12V motor.
Your question about the relay is pretty much its own answer, if you want to keep all car noise away from the Arduino, a relay is pretty much the most affordable way of changing your I/O voltage side to the "12V" pump side.
Other than those answers to what you ask, the suggestions given about do lots of research before you start building are absolutely spot on.