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Arbitrary radius - just scale up and/or down as required.

If lines are considered cylinders, they would have a radius of zero and cannot touch without intersecting.

Intuitively, it seems that radius/length ratio is what's important here.

Yes, length to radius is important factor. Here's a discussion on 7 cylinders touching (using their end) ...

No, as this cylinders can be infinitely long if I'm not mistaken

Sure, there's no lower bound on the radius/length ratio.

But there is an upper bound (think huge radius, short length).

Yes - currently the best result for coins is 5 mutually touching. Neither beaten, nor proven to be maximal.