If the coils touch other, the motion will be interfered with and it will no longer be SHM.
In gravity, the same force (the weight) is always in the same direction and doesn't affect the dynamic motion.
The sine function is just a way of describing a bit of geometry (at its simplest). You can relate SHM to circular motion in simple terms, as with the motion of a piston resulting from a crank going round, but, as a general principle, you can't do advanced Physics without bringing in Maths functions - the problems just can't be stated in a way that can be solved if you don't use some form of Maths. Newton, for instance, had to make up his own form of calculus before he could get anywhere.
The sine function is the solution
to an equation - there isn't 'an easier one'.