κορίτσι refers to a female human ("girl") but is grammatically neuter.
Grammatical gender and natural gender do not always match up.
Similarly, αγόρι is grammatically neuter but refers to a male human ("boy").
You use μια/μία when the word is grammatically feminine, regardless of whether the concept behind the word is a female human.
So you have μια εκλογή "a choice", for example -- choices are neither male nor female, but the word is feminine.
Fortunately in Greek, you can usually tell the gender of a word by looking at its ending:
- masculine: -ος -ας -ης
- feminine: -α -η
- neuter: -ο -ι -μα
But it's not 100% certain (especially -ος which can be any gender but is most often masculine), so it's best to learn the gender together with the word.
"Μία" is feminine, like "una" in Italian, or "une" in French. It means both "a" (for feminine nouns) and "one" (again for feminine nouns). As a native speaker, I would argue that "μία" and "μια" can be used interchangeably. However, I'm not aware of any grammar rules that corroborate this; I'm only speaking from experience.
It's your first example: "ego immeemiajinneeka".
Both αι and αί produce the same sound as ε, notice it is present in γυναίκα too. However in άι the α and ι are independent again e.g. γάιδαρος
For listening examples, the Greek TTS in Google Translate is generally pretty good. If you click the speaker button it will play the audio at normal speed, click it twice and it will play again slowly, and a third time will be at normal speed again. To slow it down further you could put a full stop after each word like this:
You might also like Forvo, where people upload audio recordings of themselves saying various words and phrases (just be sure not to get the Ancient Greek and Modern Greek recordings mixed up):