I was reading Mechanics by Landau and Lifshitz and I am confused when it is stated in chapter 2 section 6 that one of the integrals of motion is not independent and can be considered an additive constant of time. Hence I tried searching it up a few SE posts.

Constants of motion vs. integrals of motion vs. first integrals

According to the OP in the link above (first paragraph second sentence), we need to specify $2N$ initial conditions, one of them is the initial time, the others the initial positions and velocity.

However, shouldn't it be the initial $N$ position and $N$ velocities? Can it be shown to be equivalent? Plus aren't we working with an autonomous system of equations? Is this why we need $t - t_0$ so that it invariant to translation in time?

https://physics.stackexchange.com/a/592205/259297

The answer provided above is interesting. However, there are several points that I would like to verify...

Questions:

1. Because the Lagrangian, $\mathcal{L} (q, \dot q)$ is independent of the acceleration, Lagrange's equations, $$\frac{d}{dt} \frac{\partial \mathcal{L} }{\partial \dot q} - \frac{\partial \mathcal{L}}{\partial q}$$ which only involves one time derivative, only introduces terms linear in $\ddot q$, right?
2. According to the author (see a comment below the post as well), $$\ddot q_i =\frac{\text{d}\dot q_i}{\text{d} q_1} \frac{\text{d}q_1}{\text{d}t} =\dot q_1\frac{\text{d}\dot q_i}{\text{d}q_1}$$ How does a total time derivative become a total derivative in $q_1$? Are we performing a change of variables by inverting $q_1(t)$ to get time as a function of $q_1$ then all coordinates become $q_i(t(q_1))$?
3. Are there other proofs of this?