There are certainly people who study alternative (non-General Relativistic) theories of gravity. The most popular theories have so far been:
- Modified Newtonian Dynamics (MOND) - which essentially postulates that Newtonian Mechanics break down on some scale, leading to the rotation curves we see in galaxies.
- Tensor–vector–scalar gravity (TeVeS) - this is a relativistic generalization of MOND.
- Falling into the MOND category is f(R) gravity, which alters the general relativistic expression relating the curvature of space-time to the stress-energy tensor (the 'source' of gravity - like mass and pressure). These theories relate different functions of the Ricci scalar to this stress-energy tensor.
TeVes seems to be the most promising alternative to General Relativity (it may be able to explain things like rotation curves, predicts gravitational lensing, and some other things), but there are still many problems it doesn't do well with. So far it seems like modified theories of gravity are creating more problems than they are fixing - though I must admit, the existence of things like dark matter and dark energy are non-trivial issues cosmologists must come to terms with.
When it comes to things like alternative universes and the like as the cause of our current mysteries, I'm sure these things have been proposed. The problem here is coming up with an experiment to test them. Dark matter has been named as such because we know that it doesn't emit or absorb any radiation. As far as detecting such a particle, many real experiments have been conducted and will be conducted, and is an exciting field of research.
Below shows a few of these experiments and the range of cross-section/mass parameter space it has/will probe: