This is just the first term for a Taylor expansion at $v_2$ for small $2\bf{v \cdot} \bf{\epsilon} + \epsilon^2$ of the function $$L\left({v^\prime}^2\right) = L\left(v^2 + 2\bf{v \cdot} \bf{\epsilon} + \epsilon^2\right)≈ L\left(v^2\right) + \frac{\partial L}{\partial v^2}2~\bf{v \cdot} \epsilon$$ where only the lowest order (linear) terms in $\epsilon$ are retained. This means that in the first order Taylor term $\epsilon^2$ was neglected compared to $2\bf{v \cdot}\bf{\epsilon}$. With regard to the second part of your question $\vec v=d \vec r/dt$. Therefore total time derivative.

This is just the first term for a Taylor expansion at $v_2$ for small $2\bf{v \cdot} \bf{\epsilon} + \epsilon^2$ of the function $$L\left({v^\prime}^2\right) = L\left(v^2 + 2\bf{v \cdot} \bf{\epsilon} + \epsilon^2\right)≈ L\left(v^2\right) + \frac{\partial L}{\partial v^2}2~\bf{v \cdot} \epsilon$$ where only the lowest order (linear) terms in $\epsilon$ are retained. This means that in the first order Taylor term $\epsilon^2$ was neglected compared to $2\bf{v \cdot}\bf{\epsilon}$. With regard to the second part of your question $\mathbf v=d \mathbf r/dt$. Therefore total time derivative.