Understanding DIQKD protocol, a few questions
All of these are defined in page 2 of that paper.

1) A round is a single interaction with the device: Alice gives it an input, Bob gives is  an input, and the device outputs a bit to Alice and a bit to Bob.

2) While Alice's input is from the set $\{0,1,2\}$, the output she gets from the device, $a_i$, is a single bit. Then, the equation $a_i \oplus b_i = x_i \wedge y_i$ is on bits (the case of $x_i=2$ is treated separately). This equation is indeed the CHSH condition. If the bits where classical, the expected CHSH term is bounded by $2$, while if the system is quantum, values of up to $2\sqrt2$ are possible. By measuring the amount of times the equation holds you can estimate the value of the expected CHSH term, and if it is large enough, this implies the generation of "true" randomness.

There is a longer and more detailed version of that paper in 
https://arxiv.org/abs/1210.1810