Maybe. Some defined antibodies are associated with paraneoplastic syndromes (note: plural). In those cases the testing may be positive. However, a person could have a paraneoplastic syndrome with negative testing for the known paraneoplastic antibodies. It is possible that studies are being done to characterize tests for paraneoplastic syndromes. That might be experimental.
No. Tests for the various autoantibodies such as anti-Hu are well-established.