Assume we have a standard Brownian motion $W_t$ as the solution to the following SDE
$dX_t=\mu dt+\varepsilon dW_t$
Which kind of SDE it is? Ito process? For which kind of processes we can say that the drift coefficient and $\varepsilon$ does not depend on time or $X$? I assume that in the standard Brownian motion the increments are independent and do not depend on time. So, why in the literature mostly the coefficients depend on time and $X$?
How can i relate this SDE to the Langevin equation governed on the movement of a massless particle immersed in a flow(Brownian motion)
$0=-\gamma \dot x+f(t)$
Your question is rather unclear. First, the Brownian motion doesn't solve $$\,\mathrm d X_t=\mu\,\mathrm d t+\varepsilon \,\mathrm d W_t,\quad \mu\in\mathbb R, \varepsilon >0.$$ A solution of such equation is called Brownian motion with drift. Moreover, you can find a closed form of the solution : typically $$X_t=X_0+\mu t+\varepsilon W_t,\quad t\geq 0.$$
Notice that a stochastic process is an "Itô process" if and only if it solve an SDE. But an SDE is not an Itô process.