Alternate method to derive drift velocity:
Consider a field $\vec{E}$ inside the conductor.
Using equations of motion we can say that for every charge inside the conductor, $$\vec{v_1}=\vec{u_1}+\frac{\vec{E}e}{m}t_1$$ $$\vec{v_2}=\vec{u_2}+\frac{\vec{E}e}{m}t_2$$ $$.$$ $$.$$ $$.$$ $$\vec{v_n}=\vec{u_n}+\frac{\vec{E}e}{m}t_n$$
where $t_1,t_2,...t_n$ are the times until each of them collide with another particle respectively.
Summing them and dividing by the number of charge particles($N$) we get, $$\sum_{i=1}^n\frac{\vec{v_i}}{N}=\sum_{i=1}^n\frac{\vec{u_i}}{N}+\frac{\vec{E}e}{m}\sum_{i=1}^n \frac {t_i}{N}$$
We can say that $$\sum_{i=1}^n\vec{u_i}=0$$ as the charges initially are in perfect random motion. Thus substituting $\sum_{i=1}^n\frac{\vec{v_i}}{N}$ as $\vec{v_d}$, the drift velocity and $\sum_{i=1}^n \frac {t_i}{N}$ as $\tau$, the relaxation time for each particle we get
$$\vec{v_d} = \frac{\vec{E}e}{m}\tau$$
I don't know your course, but I can say a few things that may be of help.
You ask :"But why are we sure that when the first collisions happen, the electrons are decelerated exactly as much as the acceleration caused by the field, so it cancels out (it just balances, as the yellow marked text says)?"
The collision with an ion doesn't necessarily stop an electron, just kicks it in some random direction, s.t. the electron doesn't move anymore in the direction of the field.
But after being kicked away, the electron is again accelerated in the field direction for a time which is on average $\tau$. And so on, an almost periodic movement. Averaging on this movement, and over all the electrons, one gets the drift velocity that you say, for a given strength of the field $\vec E$.
With increasing the field, the velocity acquired by the electrons during their free flight increases, which in turn decreases the cross section of collision with an ion. So, the drift velocity should therefore increase also.
Best Answer
The acceleration only depends on the applied Electric Field (because of the potential difference created by the battery). Its value would be $\dfrac{charge*Electric Field}{mass}$
The relaxation time is just the average value. Collision time is related to kinetic energy so it would depend on the temperature. Hence, constant.
See this for more on relaxation time.
Average values are used in this equation. The motion of the electrons in the absence of an electric field will be equally distributed in all the directions, making the net value of initial velocity $u = zero$.