Could it be said just ταχυδρομική or it is necessary to have κάρτα?
You can just say κάρτα (but this is any card). Ταχυδρομική cannot be used on its own with that meaning.
Why are Greek words, in general, so long? I know 3 languages and none of them have words as long as the ones in Greek. What could be the reason for that?
A lot of Greek words, like ταχυδρομικός, are compound words. Usually, there is the word root, which may or may not be a compound, and the ending which specifies what type of word it is (e.g. a verb, a noun indicating location/gender/other meanings, plus conjugation/declension forms etc.). Ταχυδρομικός is formed of ταχύς (fast) + δρόμος (road) + -ικός (adjective ending). Ταχυδρομείο (post office) has the -είο ending signifying location. Sometimes there are prefixes too.
You can read more on how these grammatical phenomena affect word length here: https://en.wikipedia.org/wiki/Word_formation.
When you listen to tge vouce at normal speed it is difficult to hear the η in Αυτή η ταχυδρομική
It's clearer on tge slow playback
But ive tried to hear it four times on tge normal speed recording.
Is it really there?