I will try to keep this very generic, yet cover the points, mostly. My comments are in 'RED'.
"Is there any noticeable reason for the different shapes and grains of bullets? Yes.
I have been reloading/shooting 115gr plated RN bullets.
Is there any benefits to shooting 124gr FN, or 147gr? Generally, lighter bullet weight will provide a higher muzzle velocity and energy over a heavier bullet weight. However, more bullet weight normally translates to greater mass. Greater mass will provided greater momentum. So what? A heavy bullet will have a slower muzzle velocity but the greater momentum will maintain greater amounts of velocity to longer ranges and will over take the lighter and faster at the muzzle bullet's velocity at some point. In short, heavy bullets will shoot farther than light bullets.
Seems line the FN would be less aerodynamic than a RN, so why use the FN? Pointy bullets will have less drag, both in the air and in the tissues of a target. A pointy bullet that slips in and out of your target, 'paper', make no difference. But that rabbit or deer or bear, it makes a lot of difference. We want as much energy transfer into the game animal as possible. And as stated above, square edges make for nice round holes in paper for scoring.
My reasoning is, why spend more money on higher gr bullets, unless they are more accurate, but don't understand why the FN would be. Bullets vary greatly, as much as barrels, but not as much as differing loads (just too many options). Your weapon/s will be the end determining factor in this. If you pet weapon groups best with one bullet weight/type/loading, that's what you should be shooting. If not, change the many variables to find what works. Fortunately, those who came before us have tested and re-tested and general 'works better' combinations are general knowledge. Try what others have found that works but don't get stuck on just one, try several.
Also , why is there some places that sell 115gr .355, and there are one that sell .356? I have seen .355 & .356 for both coated and plated." Just more variables. Lead bullets generally will work better when over the true size of the bore by .002th, give or take. Here comes one of the biggest spreads. From the bullet weight and types listed, I am guessing the weapon is a 9MM. How many different standards, ie. bore diameters, have there been in the last 110 to 120 years for the 9MM? Add to this the sometimes sloppy standards of war production and the need or desire for several bullet diameters come into play.
Further, the normal pistol ranges are too short for 'ballistic coefficient' of the bullet to come into much play. Yes, Ed McGivern was shooting 1000 yards with his .38 Special in the 1930's, but he and Elmer Keith were exceptions.
It comes down to two things. What are you shooting, target wise, animal or paper and how much are you willing to spend on each shot. Your option.
Load with care and enjoy,
OSOK