Apart from the intrusive
H in Anthony
, these are all regular developments. There was no element of caprice
, and not much deliberate alteration
, in any of them.
For about eight hundred years no-one but Arabs and a few Irish monks knew Ancient Greek. (The Greeks of course used the then-current form of Greek, not the ancient one preserved in manuscripts.) For the rest of us, all Greek authors were known via Latin translations. The Greek Homêros is a second-declension masculine noun, and -os is the nominative case ending. The corresponding Latin ending is -us. It vastly simplified writing Latin if you could inflect borrowed words in a Latin way. For this same reason we talk about Odysseus's travels, with English possessive -'s, and don't have to worry about what the Greek genitive of Odysseus is. So, to conclude, Homêros was taken into Latin as Homerus. There were similar minor changes in other Greek names.
Ilias is the nominative in Greek: the stem is Iliad-, as in the genitive Iliados. Actually the nominative is this plus a nominative ending -s: but very early in Greek (long before the classical period) a phonetic simplification changed iliads into ilias. (Actually wiliads into wilias, but the W doesn't appear in our story.)
We have now reduced the problem to why Latin names lose endings when taken into English. We can ignore Greek from now on.
The answer to this is that it's the same reason that modern Latin speakers speak Latin differently depending on whether they live in Rome, Paris, or Madrid. In fact they even call the Latin they're speaking by different names: Roman Latin-speakers call it italiano, Parisians call it français, and Madrileñas call it español. The language has changed continuously, smoothly, invisibly, as it's been passed from parent to child. No choices or intentions were involved; no abrupt alterations occurred.
Latin lost its case marking over the centuries after the classical period. It also lost some of its sounds, and others changed. The -um of the accusative (Marcum Antonium) and the -ô of the ablative (Marco Antonio) both changed to -o, which is now the normal ending in Spanish, Portuguese, and Italian. The nominative was lost entirely. This is why the non-nominative stem Iliad- has survived.
In French, or more accurately in the Gaulish Latin that later gave rise to French, the change continued. The -o was lost. So the names Homero and Marco Antonio became Homer and Marc Antoni. It was at this stage that the Norman Conquest took place, and these French names were borrowed without change into English.
In Old English only C was used, not (normally) K. With the influx of Norman French words in which CI and CE were pronounced SI and SE, it became necessary to devise a new spelling for native words with KE and KI, like kind, king, kitchen. At some point -- and I don't know the reason -- the convention arose of indicating a K sound after another consonant by K: sick, lock, mark, work, task, risk, ink, bank.
It might have been for scribal clarity, as the other change was: an I at the end of a word was written Y. So happy ~ happier. This, put together, is how we get Mark Antony as the perfectly regular reflex of Marcus Antonius. The process continues. Starting about two hundred years ago, many dialects of English lost the R. We don't write Maak Antony, those of us who say it, but it's just another regular, unconscious change.
The equivalence of I and Y was possible because by then they represented the same sound. In fact they had since Classical Latin days, since the Greek Y didn't occur in Latin and was pronounced (by the common folk) as an I. Likewise, the Greek KH PH TH sounds, though distinct from K P T in Greek, were not distinguished in Latin. This is why T and TH could be used for the same sound, the T in Antony/Anthony. (Of course a recent phenomenon is to alter the pronunciation of this TH as if it was the thin sound.)
Ulysses was pronounced something like OO-liss-ayss in Latin, the way James Joyce said it, by the way. It's changed to yoo-LISS-eez in modern English, also for many and varied but always regular reasons. That we don't record them in the spelling is just a historical quirk. It is actually the Latin name that requires explanation here, not the English one. They also called him Ulixes. Neither of these makes sense in terms of known rules of borrowing from Greek into Latin. Admittedly, in Old Latin, the phase before Classical Latin, there was a change D > L under some circumstances (e.g. dingua 'tongue' became lingua), the Sabine L as it's known. But there is still no good explanation of the origin of the U or the X.
Now Ulysses (Latin Ulixes, actually) is far more ancient than the Greeks and Romans: he and Sinbad and Noah and Deucalion go back to Ziusudra and Gilgamesh. Various claims have been made as to how Ulixes/Odysseus could derive from older names and tongues through sound changes now too obscure to understand: a Hittite name Ulliyash is a possibility. Another that was mooted after the decipherment of Linear B and analysis of how personal names were formed in Mycenaean is that it is a regular Greek form, from O-dukj-eus 'one who leads on'.
I don't know the answer there.