Objective reduction

Yes, you can look up Diosi-Penrose gravitational state reduction. Other objective collapse approaches are Ghirimi-Rimini-Weber and Continuous Spontaneous Localization. It is not clear if these theories can reproduce the full range of quantum phenomena, but here are some references:

http://arxiv.org/abs/1410.0270
http://arxiv.org/abs/1402.5421
http://arxiv.org/abs/1209.5082

 

Regarding relativity, the issue is subtle, so I don't know if this is exactly right. Also, terminology varies, for example "causality" is sometimes taken to mean no superluminal siganalling, and at other times it is taken to mean relativistic causal structure.

First, surprisingly, relativity itself permits nonlocality in the sense that from an operational point of view, a theory can be viable for making predictions as long as it does not allow you to signal faster than light. In fact, the constraint of no faster than light signalling allows more nonlocality than is present in quantum mechanics: http://arxiv.org/abs/quant-ph/9709026. So if we take the wave function in quantum theory as real (FAPP), then the collapse is clearly nonlocal. However, the collapse does not allow faster than light signalling of classical information, so quantum theory is viable as a relativistic theory, eg. http://arxiv.org/abs/1007.3977.

So nonlocality and relativity are compatible. What about the Bell inequalities then? There it is relativistic causal structure that is ruled out - no theory that respects relativistic causal structure can explain the nonlocal correlations of quantum mechanics. So relativistic causal structure is a tighter requirement than no faster than light signalling of classical information.

How about Lorentz covariance - can we have a theory that is nonlocal, lacks relativistic causal structure, does not allow faster than light signalling, and is also Lorentz covariant? I don't think there is anything that rules that out, but I don't know how far such a theory can be taken. The issue is discussed in eg. http://arxiv.org/abs/1111.1425 and http://arxiv.org/abs/1412.6723.

 

Tricky, tricky question. The papers by Bedingham et al (2011) and Pearle et al (2014) in post #3 address the question. I don't know the answer, would love to see those papers discussed.