Does the christian god have a name?
Well there are a number of names: El, El-shaddai (Hebrew), The Ancient of Days (Hebrew origin, I think), the First cause, the Unmoved Mover (Greek Philosophy), Arche Anarchos (Patristic Tradition), Lord, Father, I am ... it depends on who you ask.
Or is he always referred to as god, heavenly father, etc.?
Father is the best way ... Abba
I know it's a queer thought, but isn't it a bit sad to be called by the role, though?
Depends on the role? The term 'father' designates a relation, not a role. My kids (who call me dad) once asked how I felt about being called Tom. I said Tom is nothing special, anyone can call me Tom, but only my kids can call me dad ... that's something special.
I guess christians consider him as the only ultimate one, not requiring to be distinguished further by any name, but I wonder if god himself feels sad to be called by the role by everyone all the time.
Another good reason for no names is to avoid over-anthropomorphisation ...
Maybe the faith can get more personal if god could be called by some name.
No name more personal than Abba.
Re your later point on 'jealous God' and so on ... there's quite a detailed explanation of the genres of Scripture in the (Catholic) Christian Tradition. Also Dionysius the pseudoAreopagite (St Denys in the East) explains in detail how such predications are indicative, and not determinative. The words 'jealous God' should stop you and make you think: "How can God be jealous?" Answer: "God can't ... so what is Scripture trying to tell me?"
There is also a theological view that the name given to Moses on Sinai "I am that I am" is not a name, but a roasting! They have followed me this far, and now they're asking my name? And how the heck do they think I have a name? Can they comprehend the mind of God, then what use is a name? ... so there is an exegesis that 'I am that I am' is a designation, that God is, whereas everything else is contingent ...
The words of Christ to St Catherine of Sienna give us something to think about: "I am He Wo Is, you are she who is not... "
God bless,
TomD