Sorting algorithm, such that each element is compared $O(\log n)$ times, and doesn't depend on a sorting network
Upon discussing this with Michael T. Goodrich, it seems that the parallel sorting algorithm by Cole for EREW PRAM does the job. See


Richard Cole: "Parallel Merge Sort". SIAM J. Comput. 17(4): 770-785 (1988).


In that algorithm there are $O(\log n)$ rounds and in each round each element participates in $O(1)$ comparisons. (One has to understand the algorithm to see that we do not abuse of making copies of each element.)

An extension of that algorithm for parallel pointer machine is given in 


M.T. Goodrich, S.R. Kosaraju: "Sorting on a Parallel Pointer Machine with Applications to Set Expression Evaluation". J. ACM 43(2): 331-361 (1996).