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What is the most appropriate dimension (e.g. 20mm x 3.33mm, 5N/mm) of the metal spring for this LEGO MOC Simple Suspension so that it will not cause damage to the LEGO part surrounding it?

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The inner space of the spring will be fine if you use an axle (4.8mm)

The outer space of the spring will be fine at 1L (7.8mm) and shouldn't rub against anything.

Here is a mockup of the spacing between everything, as you can see there is ample space between the spring and the orange pin joiners:
enter image description here

This shouldn't be a problem in your case but note that if the axle is rotating and the spring is compressed and is against a pinhole, the groove of that pinhole can cause the spring to uncoil and be damaged.

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