1a) Go back to the definition of Nash equilibrium. If p1 chooses 1, p2 should choose D. If p1 chooses 2, p2 should choose A. 3:C 4:A.
If p2 chooses A, p1 should choose 2. B:1 C:4 D:3.
There is a match: p1 selects 2, p2 selects A. This is the only match, so there is one Nash equilibrium.
1b) Strictly dominated, definition again. No matter what p1 chooses, p2 will have a higher return by choosing C rather than B, so in all cases C is superior to B. Thus, let us eliminate B from the possible options.
At this point, does either player have any strategies which are strictly dominated by others? Observe that p1 will now pick strategy 2 over 1, regardless of what p2 chooses to do. Thus, let us eliminate 1 from the possible options as well.
Repeat as necessary until the game is either solved or unsolvable. (Answer: Eliminate D, 3, C, 4 in that order.)
Not going to do Q2. Part (a) is literally copy-paste from your text, and I've already explained the idea behind (b).