Same as NL -- but if there's an oracle, then NLOG can make queries nondeterministically on a polynomial-size, one-way oracle tape. (NL, by contrast, can use nondeterministic transitions only on the worktape; oracle queries have to be deterministic.)
See [LL76] or [HCK+88] for more information.
Although NLOG is contained in P, there exists an oracle relative to which that is not the case. This illustrates that care is needed when defining oracle access mechanisms.
No class.