As long as the Stardust Dragon was summoned properly (ie, it reached the field correctly initially, rather than someone tried to summon it, and the summon was negated) then yes, you can resummon it with cards like Call of the Haunted. That applies to all Synchro Monsters, unless the monster has a specific restriction on it.
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If you use Stardust's effect, then it is tributed but not destroyed. That means a Call of the Haunted that summoned it, will remain on the field. However since Equip Spell cards can't remain on the field without a target, Premature Burial will be destroyed.
When Stardust Dragon uses its effect, it is 'tributed', which is not a destruction effect. Because it leaves the field, it means Premature Burial loses its target, so Premature Burial will go to the graveyard too. It will not inflict a destruction effect on Stardust Dragon.
There are 2 ways to get Stardust. One is to unlock the pack "The Duelist Genesis", which can be obtained by defeating the following duelists 3 times each: Barclay, Shuji, Taniyama, Ellison Noula, Skathi, Yagishita, Geruzet, Motoi, Kyono, Fudagawa, Dio, and Vega. Stardust Dragon appears as an Ultra Rare in that pack. The other way is to use the card password for Stardust Dragon (44508094) but only after you've unlocked the card, meaning you'll need either a high completion percentage of cards on the Duelist Genesis pack or to have previously obtained Stardust Dragon in the Duelist Genesis pack. Of course you could always cheat. :P
Yes you can.
No. Judgment Dragon says it 'cannot be special summoned except by...' and this way is the only way it can ever be special summoned. It cannot ever be revived from the graveyard by cards like Monster Reborn.