OK, I think found (most of) the answer to this.
According to Kallenberg (p. 451, Theorem 23.20), by the Rao Decomposition Theorem, is (up to a constant) a local quasimartingale if and only if it is a special semimartingale.
(More information about quasimartingales and the Rao decomposition theorem:
It doesn't say this in Kallenberg, but presumably any quasimartingale is also a local quasimartingale so quasimartingales are in effect only a special subclass of a special subclass of semimartingales.
Since supermartingales are the analog of decreasing functions, and quasimartingales (by the Rao Decomposition Theorem) are always the difference of two non-negative supermartingales, they are somewhat the martingale/stochastic process analog of functions of bounded variation.
Special semimartingales (i.e. local quasimartingales) are more general, in that they admit decompositions as M+A, where A is of locally integrable variation (hence of locally finite variation) and predictable.
Assuming that the conclusions of the Doob decomposition theorem (discrete time case) carry over to the Doob-Meyer decomposition theorem (continuous time case), then we also have the decomposition of quasimartingales as M+A, where M is a martingale, and A is a predictable process of finite variation. (I think.) So quasimartingales are the smallest generalization possible that encompasses but is strictly larger than submartingales and supermartingales. (Maybe.) Why they are never discussed in the discrete time case is beyond me however.
This would also seem to imply that quasimartingales immediately have discrete time analogs. For more general stochastic processes (i.e. local quasimartingales/special semimartngales and semimartingales), we automatically require the notion of local martingale. Hence whether or not those two notions possess discrete time analogs is contingent upon whether or not the notion of local martingale does.
It is not immediately clear to me whether localization of properties can only be well-defined in the continuous time case. Hence I am still uncertain about whether or not local martingales, local quasimartingales, or semimartingales possess discrete time analogs.
To clarify, Kallenberg states that (p. 436) a semi-martingale is a right-continuous, adapted process which admits a decomposition (up to constants) M+A as a local martingale and A is a process of locally finite variation.
I don't really understand the notion of localization yet, but it seems like semimartingale is the most general notion in this direction possible, given that a "local semimartingale" would locally be the decomposition of a local martingale and a process of locally finite variation; any concept requiring two iterations of localization seems likely to be redundant and unlikely to be useful.
Also I think semimartingales might be the most general possible class of stochastic integrators so in this respect probably also don't admit further generalization. They are also generalizations of Levy processes, and invariant under absolutely continuous changes of probability measure.
Hence I think I have answered 1, 2 is that they are all local martingales plus (up to constants) some process of locally finite variation (which is most cases is even predictable, provided it is also locally integrable and hence admits a compensator - which is a Levy process). 3 is also a yes, and 4 is a qualified maybe, although almost certainly a yes for quasimartingales.