BP can't rely on the Limitation of Liability Act of 1851 act. First of all, they caused the accident (I think this is what "privity" means in this context). But more important, the 1990 Oil Pollution act supercedes the 1851 act in this situation.
Yes, and no. As I understand it the Oil Pollution Act applies to owners of the vessel or those who contract the vessel, in this case the oil rig. The Limitation of Liability Act applies to the operators of the vessel. Trasocean was contracted to BP to do the drilling, operate the rig.
The "privity" section is a determination of fault. In other words the limit to liability attaches only if the operators of the vessel were not at fault. Which is a judgment call because while they were hands-on at the time they were under close supervision of BP and dependent on Haliberton's concreting job. One estimate of odds that the judgment will favor Transocean, that the Limit of Liability Act does in fact limit their liability, is roughly 50-50.