This web page use unusual variable names, which do not match the names you are used to see in other documents
in the calculator the curve equation is (Y^2 == X^3 + AX + B) mod p
(many documents use a and b in lowercase)
in the calculator n is any number
(many documents use k, while n is usually the reserved name for the order of a point G chosen as group generator)
in the calculator, if you click on nP, the calculator will calculate the addition P + P + P... + P n times, which is the multiplication of the point P by the integer scalar n (this is the ECC point multiplication)
( many documents use the notation [k]P )
if you click on P+Q, the calculator will calculate the addition P + Q following the rules that define the addition of points over an elliptic curve.
(many documents use the notation R = P + Q to design the simple addition of points (Rx,Ry) = (Px, Py) + (Qx, Qy) ). here, R and P and Q are any names, not names of reserved points for any usage.
You might find the calculator usage misleading, because it re-computes one of the curve parameters based on the points (x,y) you provide b = y^2 - x^3 - a^x . The normal usage is to fix the curve, and ensure the point (x,y) is on the curve by verifying the curve equation.
a,b,p,G,n,h are named domain parameters :
a,b,p define a curve (y^2 == x^3+a^x +b) mod p
G is a point of order n and by definition [n]G = Point_at_infinity
h is often 1 (ratio between the number of points on the curve and the number n, prime)
ECC is not generally used for encryption, but rather for key exchange ( http://en.wikipedia.org/wiki/Elliptic_curve_Diffie%E2%80%93Hellman ) and signatures ( http://en.wikipedia.org/wiki/Elliptic_Curve_Digital_Signature_Algorithm ) . Look at the wiki pages about the public keys pairs (private, public) used for ECDH and ECDSA. A private key is a big integer d, and a public key is the result of the ECC point multiplication [d]G .
if you really want to encrypt using elliptic curves, this is already answered ElGamal with elliptic curves
I am not sure about the last question ? The "compact" form of a point is the encoding of the x coordinate , knowing that y = +/- sqrt(x^3 + ax^2 +b) % p can be recomputed from the x coordinate and a indication about which of y and p-y is the right solution. Modular square root can be computed using Shanks Tonelli algorithm ( http://en.wikipedia.org/wiki/Tonelli%E2%80%93Shanks_algorithm ).
Your link to ECDRBG is a link to a bit generator which implementation is considered by the community as unsecure.