To answer your first question, a reducibility fitting your needs is log-lin-reducibility, which is logspace reducibility with the additional constraint that the size of the output string of the reduction is at most linear in the size of the input. If I remember correctly, the membership problem for context-sensitive grammars (or, if you like, linearly bounded automata) is the canonical CSL-complete problem w.r.t. log-lin reducibility.
On the applied side, the universality problem of (ordinary) regular expressions over binary alphabet, is CSL-complete w.r.t. log-lin-reducibility. The notion and the completeness result are found in Albert R. Meyer and Larry J. Stockmeyer (SWAT 1972) also: Stockmeyer (PhD thesis, MIT 1974). For further background and similar results in that area, see also the recent survey by Holzer and Kutrib (DLT 2010).
EDIT (2017/03/06): Regarding your second question, the accepted answer to the question below cites a paper by Rounds (1973), which constructs a one-way nested stack automaton recognizing SAT. While SAT will not qualify as a "natural" CSL, it might be worth to search the literature for other examples of one-way nested stack automata or indexed grammars.
Context-sensitive grammar for SAT?