What you have done is correct although it is a little peculiar that you write the vector equations in terms of "t" but have the parametric equations in terms of "s"!
For the second part, determining where the lines intersect, you want to use different variables, say, for l1,
x= 4+ 3t
y= -7+ 2t
z= 2+ 2t
and, for l2,
x= 1+ 8s
y= -3s
z= -3- 3s
At a point of intersection the x, y, and z must be the same:
x= 4+ 3t= 1+ 8s
y= -7+ 2t= -3s
z= 2+ 2t= -3- 3s.
That is three equations in only two unknowns so you can solve the first two equations for s and t and then see if those values satisfy the third one. That is because, in three dimensions, two line do NOT in general intersect.