First this hadith:
"The part of an Izar which hangs below the ankles is in the Fire." مَا أَسْفَلَ مِنَ الْكَعْبَيْنِ مِنَ الإِزَارِ فَفِي النَّارِ
is speaking about clothes such as trousers, dejllabas, the covers worn by pilgrims (for hajj) etc. which are long and loose types of clothing (Izar). Socks are not even meant in this hadith, see also this ahadith from sunan abi Dawod, sunan an-Nasa-i, sunan ibn Majah, al-Muwatta' and sahih al-Bukhari which should make clear that what is meant with a lower garment is either some kind of clothes which may refer to clothes which go down at least from the pelvic bone and "hang down" (it could also refer to what we may call underwear or nightgowns nowadays)!
So primarily you seem to have misunderstood the topic of the hadith.
Socks in Arabic are referred to with jawarib جَوَارِب and as we are allowed to pray and wipe over khuffayn ("leather socks") so on the whole there can't be a prohibition to wear socks (many sahaba also wiped on socks as you may read in this fatwa) as one prays with the kuffayn after wiping over them. Wipping on socks is accepted by some madhabs in case of thick socks (as far as I understand the shafi'i and maliki madhab wouldn't or at least shouldn't accept it because of their definition of khuffayn and explanation of the reason on allowing wiping over them).
The only situation where you shouldn't wear socks is during Ehram (Hajj/'Umrah) but if you need them (because it is cold) you may perform fidya later.
Also read this fatwa (only available in Arabic, French, Romanian and Chinese) about the the meaning of the hadith quoted in your Question post.