How many pure strategies does the player have? How many Nash equilibria? 
Consider the extensive form game between Player 1 and Player 2. 
At each decision node one player has to decide whether to stay in (I) or to leave (O). First Player 1 moves. If Player 1 leaves, the game ends. If Player one stays then Player 2 moves, and so on and so forth until someone leaves or Player 1 chooses a second time when the game ends. 
How many pure strategies does player 1 have? 
I said that player one has 4 pure strategies because they can choose between I and O in round 1 and then again they choose between I and O in round 3. 
How many pure strategy Nash equilibrium are there? 
I said the game has 3 pure Nash equilibrium at {4,0} {1,1} and {1,1}.
Can someone please tell me if I am correct with my answers? Thanks!

Player 1 has four strategies and Player 2 has only two: thus $S_1 = \{ ii, io, oi, oo \}$ and $S_2 = \{i, o\}$. 
The strategic form for this game is
$$\begin{array}{c|c|c|}
& i & o \\ \hline
ii & \underline{4},\underline{0} & -1,-1 \\ \hline
io & 3,\underline{2} & -1,-1 \\ \hline
oi & 1,\underline{1} & \underline{1},\underline{1} \\ \hline
oo & 1,\underline{1} & \underline{1},\underline{1} \\ \hline
\end{array}$$
where the payoffs corresponding to best replies are underlined. 
The Nash equilibria correspond to the three cells where both players are using best replies: $(ii,i); (oi,o); (oo,o)$. Your answers are correct, although it is more accurate to view the Nash equilibria as strategy profiles versus payoffs. (That is, $(ii,i)$ instead of $(4,0)$.)
