a) Suppose $1000$ boys and $1000$ girls arrived to the camp. We can split them into groups with $4$ students in each: $2$ boys and $2$ girls. Moreover, they know each other in the following way: $B_1 \leftrightarrow B_2 \leftrightarrow G_2 \leftrightarrow G_1 \leftrightarrow B_1$. One can check that all conditions are satisfied.

b) Suppose by contradiction that it is possible. Then every boy can be put into a pair with a girl if they know each other. Thus, there has to be exactly $1009$ boys and $1009$ girls. Similarly, all the boys can be divided in pairs if they know each other, so the amount of boys has to be even. That leads to a contradiction.