The trouble with this question is that there is no strictly objective answer... only time will tell. How would we ever know what there is left to be discovered in any situation? Only after making a discovery do we know that there was one to be made in the first place...
We have a Standard Model of Particle Physics that describes existing measurements very well. There are processes that are predicted by that model that have not yet been observed, and it is a very reasonable expectation (possibly shared by everybody...) that these predicted processes will be discovered in upcoming data as predicted. So there's work to do on this front.
Then there's the search for really new processes, in the sense that they are not predicted by the Standard Model of Particle Physics. Everybody's guess as to how may of these discoveries (or indeed any) will be made at the LHC. For sure it's not getting any easier.
For a particular dark matter model, there will usually be one or three particularly well-suited experiments to conduct to test it (i.e. to try and discover it, if it exists to be discovered). You mention axions in particular, there are a good number of dedicated experiments searching for those.
But even if an experiment isn't perfectly suited to do a given measurement, once the experiment exists, one should of course still analyze its data to get the most out of it and see whether one can e.g. make statements about axions using LHC data. That is indeed an active area of research too.
Finally, to address your last question, that answer can certainly be answered positively, if only because no-observations rule out parameter space where beyond-the-standard-model-theories can live, and thus still provide a bit of insight even in that case.