addition on finite elliptic curves
You are right, when you're working on a finite field, you should get integer results. So let's look at why you aren't.


The problem lies in how you handle the fractions inside the equation, e.g.
$s=\frac{y_q-y_p}{x_q-x_p}$. Elliptic curve operations make use of finite field arithmetic. Maybe it'd be easier for you to grasp if you imagined the equation to look like $s=(y_q-y_p)(x_q-x_p)^{-1}\mod p$. This means that you have to calculate the modular inverse of the denominator of your fracture and then multiply this with the nominator.


$R$ is indeed part of the curve. Recalculate it using the "technique" described above.
It's not right to see it as length. As you're reducing the result modulo a prime ($\mod p$) the information which you call "length" is lost.
Take two points that are part of an elliptic curve, e.g. $P$ and $Q$. Commutative means that $P + Q$ gives the same result as $Q + P$. However it does not mean that you can just switch the $x$- and $y$-coordinates