Well you're not quite looking at #1 correctly. If G was complete it would have 21C2 = 210 edges so it is almost complete - it's missing 2 edges. Think of the vertices that are end points of these missing edges (it could be two disjoint edges or a path of length 2 that is missing); it's not hard to show some of these must have odd degree (almost all have degree 20).
For # 2 start at the vertex of degree k and direct all of its incident edges away from it. Now look at the endpoints of those arcs (directed edges). Either they have no other incident edges in which case they have degree one or they have other incident edges in which case arbitrarily choose one and direct it away from the initial vertex of degree k. Since G is finite the process must end with k paths directed away form the initial vertex and the ends of these paths are vertices of degree 1.
These are only sketches of proofs you need to be more formal in your presentation.