For the English I would toss 'far-right' and 'far-left' into the discussion, which might correspond to maxx's #2.
>>das "Stramme" assoziiert man ja gemeinhin eher mit rechtem Gedankengut und militaristischem Gehabe
Yes, and I would say the same is true for the English. Perhaps because hard-right politicians really are hard in the sense of tough, hawkish on defense and military policy, 'hard-right' sounds more idiomatic to me than 'hard-left,' which I notice you don't cite examples for. Perhaps the same is true of 'stramm,' which makes me think of a stiff, upright military posture, snapping to attention.
Maybe there's also a difference in the modern day that wasn't there in the 20th century, because today far-left political parties in developed nations tend to be pacifist (and thus not 'hard' in the sense of military force), while in authoritarian nations, one-party states ruled by corrupt elites, 'left' doesn't really have much meaning any more, it's only 'hard' and little else.
And '-extrem' is indeed more extreme than just 'far-.' I'm sure there are discussions on 'rechtsextrem' etc. in the archive, but I would be inclined to use 'ultra-right-wing' in English, or else the noun 'extremists': 'right-wing extremists,' 'left-wing extremists,' implying those willing to use violence to further their ends.
But that's only my own hazy sense of the terms -- see if others have another perspective.