Maybe I made a mistake, but I think this can be solved really simply just by taking each option in turn, assuming it is correct and looking for an inconsistency.
1 can't be right because it both affirms and contradicts 2.
2 can't be right because it both denies and fulfils 4 (even if 'one' means 'one and only one', because we already know that 1 is false, and looking ahead we can see that 3 is also false).
3 affirms 1, which we already know to be false.
4 affirms one of 1-3, which we have determined are all false.
5 is correct, as demonstrated by our finding that 1-4 are all false.
6 is false; we would know this even if we hadn't already worked out that 5 was true, because 6 implies that 1-4 are all false, which implies that 5 is true.
1 can't be right because it both affirms and contradicts 2.
2 can't be right because it both denies and fulfils 4 (even if 'one' means 'one and only one', because we already know that 1 is false, and looking ahead we can see that 3 is also false).
3 affirms 1, which we already know to be false.
4 affirms one of 1-3, which we have determined are all false.
5 is correct, as demonstrated by our finding that 1-4 are all false.
6 is false; we would know this even if we hadn't already worked out that 5 was true, because 6 implies that 1-4 are all false, which implies that 5 is true.