All I see on the Gigas family regarding elements is "based on Ice". I know the site lists by crystal dropped, and maybe they really are based on Ice, but in my experience, they're weak to Earth and extremely strong against Thunder, thus I'd hypothesize they're Thunder-based. At least the ones in Qufim&the tower.
Even the Too Weak ones (mostly the NMs, Ogygos, Enkelados, ...) like to resist my ThunderIIIs, ThundagaIIIs ... while they rarely resist any other spell (they're Too Weak, it's normal they don't).
Maybe they're still Ice-based, like Bombs are Fire-based and weak to Fire. (I didn't say they're weak to Ice, just meant to say the element the mob is based on doesn't necessarily have to influence its elemental strengths/weaknesses)
About the Socks and the Shield : the NMs seem to drop them a lot, but I've never gotten any from the regular Gigases/Jotunn.