I learned Irish at school in Ireland and we were always taught verbs with "mé" ie. "ith mé". We were taught shortening it to "ithim" was an option but not necessary just like when we say "can not" in English and it can be shortened to "can't". Has that changed or were we just being taught a kind of text book version of the language rather than a natural fluent one. The school I went to wasn't in the Gaeltacht area.
Nope, it's not really the same as Spanish pronoun drop, since you use one form with the separate pronoun (itheann mé) and the other (ithim) without the separate pronoun.
That's really happening here is the form ithim has the pronoun inside it (that's what the -m) is.
Look at it this way: itheann doesn't tell you anything about the person who's eating. To do that, you need to add a noun (itheann Pól, "Paul eats") or an pronoun. If you're adding a first person singular pronoun, you have the option of doing is as a separate word (itheann mé) or as an incorporated pronoun (ithim).
As for other persons:
2nd singular: Only itheann tú: no incorporated forms that I know of.
3rd singular: Only itheann sé/sí/Pól: no incorporated forms.
1st plural: The incorporated form ithimid is much more common. In some places (Conemara?) they say itheann muid
2nd plural: I think there's a form like ithid used somewhere (Munster?), but only itheann sibh is common
3rd plural: Only itheann siad / itheann na páistí is possible. No incorporated forms.
The separate forms are correctly called "analytic forms", and the incorporated forms are properly called "synthetic forms".
th in Irish is not pronounced like "th" in English.
This Youtube video gives an introduction to the basics of Irish pronunciation - Sounds and Spelling of Irish / Fuaimniú agus Litriú na Gaeilge