States and Probability distributions that the 5-qubits IBM computer can produce
I just ran the first state you suggest (i.e. the GHZ state with negative phase in the Hadamard basis). Basically what I did was to write a circuit which creates that state, apply one of 5 stabilizer generators (XIXII,IXXII,IIXXI,IIXIX or -ZZZZZ) and then inverted the creation process. The circuit is shown below for the -ZZZZZ stabilizer.



There seems to be problem with the calibration or something, since I am seeing a bit flip occurring on the first qubit with roughly probability 2/3 that should not happen. The expected result should be 00000 in all cases, however 10000 seems to be twice as likely in almost all cases (prob ~0.1 vs ~0.2 over 1024 runs for each stabilizer generator).

Below are the probabilities of obtaining 00000 for each stabilizer generator over 1024 runs each:
    XIXII:0.10840
    IXXII:0.15625
    IIXXI:0.11621
    IIXIX:0.10645
    ZZZZZ:0.09375

And the probabilities of obtaining 10000 for each stabilizer generator:
    XIXII:0.21777
    IXXII:0.21875
    IIXXI:0.17871
    IIXIX:0.14648
    ZZZZZ:0.19726