1-- 'Hats' is preferred for consistency (since there is little danger of each man wearing more than one), but the singular is not wrong.
2-- I don't think there is a rule. It seems to me that the adjective is used if it is available. 'Oar', for instance, has no adjectival form.
3-- 'Car-hire company' is much more common in BrE, according to Ms Google. 'Car-rental station' may be more European.