Ποδόσφαιρο, Football

Ποδόσφαιρο in Greek is rather easy as it is a compound word of ancient Greek words πους and σφαίρα, meaning foot and ball.

Πους related to Latin ped Old English fot and English foot.
Alternate Greek forms: πῶς (pos), Doric πός (pos), Laconic πόρ (por), Ionic κῶς (kos).
The ionic form is most interesting, I wonder if it evolved somewhat different, giving other words in the future. The island Kos (attested since the Iliad) was part of the Doric Hexapolis, it must not be related.

Σφαίρα not related, but I can't get to an etymology either.

Football in english is comprised of the same two compounds, foot and ball.
Foot (n.)
"terminal part of the leg of a vertebrate animal," Old English fot "foot," from Proto-Germanic *fot (source also of Old Frisian fot, Old Saxon fot, Old Norse fotr, Danish fod, Swedish fot, Dutch voet, Old High German fuoz, German Fuß, Gothic fotus "foot"), from PIE root *ped- (1) "a foot" (source also of Avestan pad-; Sanskrit pad-, accusative padam "foot;" Greek pos, Attic pous, genitive podos; Latin pes, genitive pedis "foot;" Lithuanian padas "sole," peda "footstep").
http://www.etymonline.com/index.php?term=foot

Now concerning ball there is a little confusion because the English linguists only go as so far as the Germanic languages are concerned, but do go to the trouble to tie it from PIE *bhel- (2) "to blow, inflate, swell" and give a bunch of derivatives: (source also of Greek phyllon "leaf," phallos "swollen penis;" Latin flos "flower," florere "to blossom, flourish," folium "leaf;" Old Prussian balsinis "cushion;" Old Norse belgr "bag, bellows;" Old English bolla "pot, cup, bowl;" Old Irish bolgaim "I swell," blath "blossom, flower," bolach "pimple," bolg "bag;" Breton bolc'h "flax pod;" Serbian buljiti "to stare, be bug-eyed;" Serbo-Croatian blazina "pillow").
http://www.etymonline.com/index.php?term=ball
http://www.etymonline.com/index.php?term=bole

But another PIE root for “to blow, inflate, swell” exists and it is *bew- which gives words in Germanic languages only, which mean "Tree trunk".
https://en.wiktionary.org/wiki/Reconstruction:Proto-Germanic/b%C5%ABl%C5%8D

They avoid the Italian and Latin influence, let alone the Greek one.  The word the Italians use is Balla, and a word in use modern Greek too μπάλα (loan from Italian). Now looking at the italian etymology, I found something very interesting, the tie to one of the Godess Athena epithets. The italian etimology stated it either originates with Germanic, or with greek βάλειν and/or πάλλας
http://www.etimo.it/?term=balla

Βάλλω is derived by linguists by a different PIE root *gʷl̥-ne-h₁- < *gʷelH-
and which in consensus does give the Latin ballō and from there the Italian ballo, meaning "a dance" or "ball"
https://en.wiktionary.org/wiki/ballo#Latin

Someone is obviously wrong, and I suspect someone oversimplified the changes words underwent, or simply was driven by a kind of wishfull thinking scientists often fall in, by predetermining the outcome of their own research, as there is no linguist who is not himself a speaker of some language, and thus himself immersed in a cultural sub-context.
I don't have the knowledge to pass judgement on who is right, I am not a linguist myself, I have only studied Communication and Philosophy, but will make a note of this and continue searching.

As for Πάλλας, this gives πάλλω (throb), but also Παλλάδα (probably meaning virgin, but may also mean pure, shinning, protector etc.), παλλάδιο (palladium), παλληκάρι (stalwart), παλλακίδα (concubine)
https://el.wiktionary.org/wiki/%CE%A0%CE%B1%CE%BB%CE%BB%CE%AC%CE%B4%CE%B1

I want to thank Chrissa Marasli for pointing out there's something going on with the etymology of the word "ball".