I read somewhere that G is the same for everyone in Bitcoin. I am guessing this means the Generator and its properties, not that it's a scalar or some type of simple constant. Like you say, G is a point on a curve, and everyone uses the same curve.
In order for the elliptic curve cryptography to work we all need to use the same Elliptic Curve. And that curve is defined by a sextuple T = (p, a, b, G, n, h), so yeah G is defined by the curve and is the same when using the same curve.
So with G point on the curve, P, n is the smallest positive integer that satisfies nP = O where O is the order.
No, O is defined as the point at infinity.
In order for the elliptic curve cryptography to work we all need to use the same Elliptic Curve. And that curve is defined by a sextuple T = (p, a, b, G, n, h), so yeah G is defined by the curve and is the same when using the same curve.
So with G point on the curve, P, n is the smallest positive integer that satisfies nP = O where O is the order.
No, O is defined as the point at infinity.
So, we are aiming for the number of times a point can be added to itself so the slope is infinite? i.e. if you looked at a simplified graph, the slope of the line is vertical, or nearly vertical. Or the order is some way you can detect the number of times a point addition can be done to itself before you hit O (infinity). Is the order selected after a number of point additions, or is there an algorithm that gives you an order after repeated point doubling? (Is that the same as adding a point to itself?)
This seems to be the same, but with or without a prefix of 04. So this is a starting point to generate other points on the curve? Oh hang on. I'll get a book on crypto. 
If you want to understand why there are two public keys (with 0x04 and 0x02/0x03) read this comment:
https://bitcointalk.org/index.php?topic=5049111.msg46800542#msg46800542
https://bitcointalk.org/index.php?topic=5049111.msg46800542#msg46800542
Thanks, that was a very accessible post!