That's how Greek works :)
Its demonstratives take a definite article -- so they say literally the equivalent of "this the apple".
Similarly with possession; we say το βιβλίο μου ("the my book" or "the book my") in Greek.
Αυτό μήλο would be simply incomplete.
If you ever take Hungarian, you will see the same thing.
No, not always.
After "this" or "that", yes, or in combination with "my/your/his/her/our/their".
But otherwise, there may be an article or there may be none.
A bit like in English where you can say "I am drinking water" (any old water) versus "I am drinking the water" (some specific water): πίνω νερό / πίνω το νερό.
The rules for using or omitting an article are not quite the same in Greek compared to English, but there doesn't always have to be an article before a noun in all cases.
How do you know when to use ι and when to use η? They both make the same sound! And then there are all the diphthongs that make the same sounds as well: ει, οι, and υι. Is there any trick or grammatical rule to this?
No - no trick or rule. Just memorisation.
It's like "ee" and "ea" in English, where you have to learn whether to write "meet" or "meat", "reed" or "read", etc.
And for the same reason: those sounds are written differently because they used to be pronounced differently, and the writing system is partly historic/etymological.
(That said, you can ignore υι since in Modern Greek it only occurs in two or three word stems such as "harpy" (the mythological animal) and "adopt".)
What can sometimes help is if the Greek word was borrowed into English -- if the English word has an "e" there, the Greek likely has an η or οι in that syllable; if it has a "y", then υ; if an "i", then ι. For example, "hymn" has a y and ύμνος has an υ; "telephone" has te- and τηλέφωνο has τη-; "cemetery" has "ce-me-te-" and κοιμητήριον has κοι-μη-τη-.