Suppose $X(t)$ is a Levy process with almost surely positive increments (for all $t_1 < t_2$ $P(X(t_1) < X(t_2)) = 1$)
Define
$$\nu X(t) := \sup \{\tau \in \mathbb{R_+}| X(\tau) < t\}$$
It is not hard to see, that $\nu X$ is also a stochastic process with almost surely positive increments.
My question is:
Is it a Levy process too?
It is not hard to see, that $\nu X (0) = 0$ and for all $t_1 < t_2$ $\nu X(t_1) - \nu X(t_2) = \mu\{\tau \in \mathbb{R}| X(\tau) \in (t_1; t_2]\}$, which means, that the conditions (1), (3) and (4) are satisfied. However, I do not know, whether the increments of $\nu X(t)$ are independent or not. They are indeed uncorrelated, but uncorrelatedness $\neq$ independence.
One can also notice, that $\nu \nu X(t) = X(t)$ almost surely. However, it does not seem to be very helpful.
No. For a counter-example just take $X(t)$ as a Poisson process.
For $t \geq 0$, let $X(t)$ count the number of arrivals in a Poisson process of rate $\lambda>0$. Note that $X(t)$ is a non-decreasing staircase function that is right-continuous. It has i.i.d. exponential inter-arrival times $\{T_i\}_{i=1}^{\infty}$ and has jumps $\{J_i\}_{i=1}^{\infty}$ of height $1$ (so $J_i=1$ for all $i$).
Now the process $Z(t) = vX(t)$ simply exchanges interarrival times and jumps: $Z(t)$ has inter-arrival times of 1, and jumps of heights $\{T_i\}_{i=1}^{\infty}$. Since $Z(t)$ has arrivals at periodic times, it does not have stationary increments (it is not a Levy process).
Also, your edit in reaction to my first comment does not fix the problem of $t=0$:
$$Z(0)=vX(0)= \sup\{\tau \in [0, \infty): X(\tau)<0\} = \sup \phi \neq 0$$ This is because there is no time $\tau \geq 0$ for which $X(\tau)<0$. So your definition of $vX(0)$ is taking a supremum over the empty set.