1) Line: x = y = z
Parametric form: x = t, y = t, z = t. L(t) = (t, t, t).
Vector form: L(t) = vt, where v = (1, 1, 1) = i + j + k.
2) I am *not* sure what "projecting a line L through a vector w onto a plane P" means. I'll give you my thoughts, but leave it to you, your book, or someone else to provide the exact meaning. Without knowing the exact meaning of what's being asked, you obviously can't solve the problem. I haven't thought this out, so it's possible that two, or even all three, of the following are the same:
Option #1: For each point on L, draw a new line between it and the line formed by the vector w through the origin, so that the new line, and the line formed by w, are perpendicular. Where that newly drawn line intersects P is the projection of that point onto P. When you do that for every single point on L, you'll have projected the line L through w onto P. (Note that this approach breaks down where those two lines intersect.)
Option #2: Intersect P with the plane formed by the line L and the vector w at the origin. The line of intersection between those two planes will be the projection of the line L through w onto P. (Note that this approach exploits that L goes through the origin.)
Option #3: Decompose w = w1 + w2, where w1 and w2 are perpendicular (i.e. orthogonal), and w1 is parallel to L. The idea here is that the w1 part of w doesn't contribute to the projection of L through w, so that "projecting L through w" should mean "take each point of L, and draw the line through it in the w2 direction, and where that line intersects P is the projection of that point through w onto P".