Ο, ο όμικρον (όmikron) from o+μικρός/ small, that is the short o in old Greek for which this unique alphabet was created: every sound in old Greek got its own letter. In order to read and write you just had to learn the 24 letters
Ω, ω ωμέγα (oméga) from o + μέγας/ big, great, that is the long-sounding o in old Greek
No, both ómicron Ο; ο and oméga Ω; ω, are pronounced like "o" in British got, a clean one-vowel sound. I said that in old Greek Ω is a long o-in-dog sound while O is short. The same for épsilon Ε; ε and Éta Η; η in old Greek. Well in ancient Greek both were the e-in-forget sound, épsilon was short and éta long.
There is considerable variation in the pronunciation of vowels among English speakers, but no English speakers pronounce the O in "dog" like the Greek O (both omicron and omega). The vowel sound in "dog" is the same as "not." It's the one represented by the alpha in this linked chart. (And in fact, that sound is closer to the Greek alpha sound than the omega/omicron sound.) The sound in Greek is represented by the omicron on this chart. https://en.m.wikipedia.org/wiki/Vowel_diagram#/media/File%3ACalifornia_English_vowel_chart.svg
The closest example I can think of would be "go" or "no," but in Greek the sound is shorter and crisper than in English.
Both όμικρον o and ομέγα ω are pronounced
«like [o] in “got” the way it is pronounced in British English. Notice how the vowel in British “got” is tense, which means that you should really round your lips when you pronounce the Greek [o]. A mid-close back rounded vowel.»
from "The Greek Alphabet" http://www.foundalis.com/lan/grkalpha.htm
which gives: letter, name, SOUND, pron. in Modern Greek, pron. in Old Greek (attic dialect)
Apparently I do not pronounce English correctly so I replaced "dog" with "got" in my earlier comment and give the above as a complement