Route to Infinity
Yes, the route will visit (18,17) - it goes through all points
(x,y) where x and y are positive integers. At some stage, the route
will pass along the diagonal corresponding to the sum of the
co-ordinates being x+y, and it hits all points on that diagonal
that have both co-ordinates positive.
At (18,17) the route will be going down and to the right, because
this is what it does on diagonals where the sum of the co-ordinates
is odd. This means that the next point visited will be
(19,16).
The route visits 74 points before reaching (9,4).
It passes through 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 =
(1/2) x 11 x 12 = 66 points that lie on diagonals completed by the
route.
The diagonal passing through (9,4) will be going from top left to
bottom right (as 9 + 4 = 13 is odd). Before it reaches (9,4), the
route will have passed through the points with sum of co-ordinates
equal to 13 where the x co-ordinate is one of 1, 2 3, 4, 5, 6, 7
and 8, so there are 8 such points.
So the route will have visited 66 + 8 = 74 points before it reaches
(9,4).