I've made some headway on Question 3. It is related to my studies of Jacobsthal's function. Although a theory for general square free numbers $Q$ can be developed, I will take $Q=P_n$, the product of the first $n$ primes, for this post.

If I restrict the algorithm to primes dividing $Q$, the union of the trajectories are those numbers not coprime to $Q$, and as the system can be modeled as a finite state automaton, the trajectory of each prime will be periodic with a period related to a multiple of $Q$. More precisely, for a given prime $p$ dividing $Q$, one has that a positive number $m$ is in the trajectory if and only if $m+Q$ is in the trajectory. (Temporarily, I include $p$ in its own trajectory, contrary to the post above.)

To see this, note that $\gcd(Q,Q+p)=p$ for every prime divisor $p$ (and also every divisor) of $Q$. Thus, modulo $Q$, there is only one way to arrange the $n$ primes at the start ( the pattern of primes assigned to the indices in $[mQ+2, mQ+p_n]$ is independent of $m$), and thus the pattern of primes assigned to c repeats modulo $Q$ as well.

It is tempting to conjecture that c[$m$]$=$c[$Q-m$] for positive $m$ less than and not coprime to $Q$, but when $Q=30$ one has c[$12$]$=2$ and c[$18$]$=3$ (and similar examples exist for some larger $Q$). It is also tempting to conjecture that each prime $p$ does slightly less than $Q/p$ jumps in a range of length $Q$, with the difference related to inclusion-exclusion. Simulations show much smaller numbers than this prediction. Note that the longest jump is bounded above by the sum of the $n$ primes, and is likely to be much less. For small values of $n$ $(n\leq 9)$ I see the largest jump as less than $3p_n$. I would not be surprised if the largest jump were of the same order as $g(Q)$, the Jacobsthal function applied to $Q$, which would be conjectured at or near $p_n(\log p_n)^2$ but I would settle for strictly less than $O(n^2)$. In any case, it is apparent that the set of primes stay in a rather tight cluster not much larger than the largest prime as they jump according to the restricted version of the algorithm. How tight is hopefully less hard than various open questions regarding the distribution of prime numbers.

Gerhard "More Jumping Means More Excitement" Paseman, 2016.09.11.

I've made some headway on Question 3. It is related to my studies of Jacobsthal's function. Although a theory for general square free numbers $Q$ can be developed, I will take $Q=P_n$, the product of the first $n$ primes, for this post. Of course, $2$ can only make a jump of size at most $2n$, but it will rarely make a jump of that size for large $n$.

If I restrict the algorithm to primes dividing $Q$, the union of the trajectories are those numbers not coprime to $Q$, and as the system can be modeled as a finite state automaton, the trajectory of each prime will be periodic with a period related to a multiple of $Q$. More precisely, for a given prime $p$ dividing $Q$, one has that a positive number $m$ is in the trajectory if and only if $m+Q$ is in the trajectory. (Temporarily, I include $p$ in its own trajectory, contrary to the post above.)

To see this, note that $\gcd(Q,Q+p)=p$ for every prime divisor $p$ (and also every divisor) of $Q$. Thus, modulo $Q$, there is only one way to arrange the $n$ primes at the start ( the pattern of primes assigned to the indices in $[mQ+2, mQ+p_n]$ is independent of $m$), and thus the pattern of primes assigned to c repeats modulo $Q$ as well.

It is tempting to conjecture that c[$m$]$=$c[$Q-m$] for positive $m$ less than and not coprime to $Q$, but when $Q=30$ one has c[$12$]$=2$ and c[$18$]$=3$ (and similar examples exist for some larger $Q$). It is also tempting to conjecture that each prime $p$ does slightly less than $Q/p$ jumps in a range of length $Q$, with the difference related to inclusion-exclusion. Simulations show much smaller numbers than this prediction. Note that the longest jump is bounded above by the sum of the $n$ primes, and is likely to be much less. For small values of $n$ $(n\leq 9)$ I see the largest jump as less than $3p_n$. I would not be surprised if the largest jump were of the same order as $g(Q)$, the Jacobsthal function applied to $Q$, which would be conjectured at or near $p_n(\log p_n)^2$ but I would settle for strictly less than $O(n^2)$. In any case, it is apparent that the set of primes stay in a rather tight cluster not much larger than the largest prime as they jump according to the restricted version of the algorithm. How tight is hopefully less hard than various open questions regarding the distribution of prime numbers.

Gerhard "More Jumping Means More Excitement" Paseman, 2016.09.11.