Basics: The Lagrangian in field theory was written as $$\frac{\partial \mathfrak{L}}{\partial \varphi}=\partial_\mu(\frac{\partial\mathfrak{L}}{\partial(\partial_\mu\varphi)})$$.

Question 1:

Is $\partial_\mu\varphi=(\partial_t+\partial_x)\varphi$ now a vector/tensor, or still a single variable?

Question 2:

Further, is there any difference between $\displaystyle \frac{\partial (\partial_\mu\varphi) }{\partial (\partial_\mu \varphi)}$ (which equals to $1$ I suppose) and $\displaystyle \frac{\partial (\partial^\mu\varphi) }{\partial (\partial_\mu \varphi)}$?

Question 3:

How to take derivative with respect to complex field? i.e. $\displaystyle \frac{\partial (\partial_\mu\varphi^\dagger) }{\partial (\partial_\mu \varphi)}$ , $\displaystyle \frac{\partial (\partial^\mu\varphi^\dagger) }{\partial (\partial_\mu \varphi)}$ and $\displaystyle \frac{\partial \varphi^\dagger}{\partial\varphi}$?