First, the floor does not need to be parallel to rail. The proper direction would be perpendicular to the joists, regardless of whether that is parallel or perpendicular to the railing.
Second, as mentioned above, it's probably best to use a pro as the cuts are tricky no matter which way you do them.
Third, not sure if there is some flexible shoe molding you can use to cover the gaps (as others mentioned it's best to have some space for expansion/contraction). When we have curved steps at the bottom, we use flexible shoe molding to hug the curve (we do the same if the wall is curved). Usually, I see this white primed. I don't know if they make them in wood or not, but maybe you could find some and then stain it to match.
Debbie Gartner aka The Flooring Girl
http://TheFlooringGirl.comWestchester County, NY