There's no difference between them, though 'nonetheless' is less common. As they are normally suprasentencial (contrasting sentences), they can be replaced by 'however', but 'although' normally contrast two clauses within the same sentence so wouldn't work as a replacement. 'Though' is used with a similar effect to however/nevertheless when it placed at the end of the second sentence, but this use is mostly found in informal writing.
1 Change the punctuation to a semi-colon or period.
3 Bit of a mess- you have both but and nevertheless in there.