Your mortal shell is mortal: your soul is that of a Dragon, with no other distinction aside the fact that it was born into a soft meatsuit rather than winged, scaled, weapon of mass destruction.
On the note of Gods, they can be killed. It's just not something that is particularly easy to do even for other Gods, nor is it something that happens very often. And even at that, the term "God" as it relates to The Elder Scrolls isn't well defined: some entities referred to as Gods are very much killable (the Tribunal), others take a bit of work (Talos), and others still are tough nuts to crack unless you yourself are a God (Daedric Princes). Parenthetical list not exhaustive, just used as example.
And Alduin is not a Demigod, he is a God. He is the World Eater, and the Nordic aspect of the Time Dragon. From an out-of-universe perspective, he predates Akatosh, and only refers to himself as firstborn by virtue of the retroactive changes Akatosh made to the timeline in order to secure his place as the most prominent aspect. There was a thread that was linked above that does well in explaining the whole situation with Aka/Aka-Tusk and his shards-that-are-aspects.