The Stardust Dragon can use its negation effect, but will not be able to resummon itself. Special summon-only monsters (which Synchro, Ritual and Fusion monsters are) must be 'properly' special summoned first (ie, Synchro, Ritual or Fusion Summoned) if you want to resummon them from the graveyard.

There are cards that can get them on the field without properly summoning them, like Cyber Stein, and Starlight Road. If you summon them with these, the monster can't later be revived from the graveyard after it gets destroyed.

Yes, you can, it is a special summon and can be responded to as normal. However it is more than likely that the monster's controller will simply tribute it to negate and destroy Bottomless Trap Hole, then resummon the dragon in the same end phase.

In your own standby phase, yeah. It is a Quickplay spell, so can be played from hand in your own turn. You can draw it, then activate it in the Standby phase, tributing Treeborn Frog, then resummon the frog before the standby phase ends.

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.

Once properly summoned, the Chaos Sorcerer will 'remember' that as long as it is never returned to hand or deck. So if you used Monster Reborn on it, and it was destroyed, you can then use Call of the Haunted on it. But if you were to have it returned to hand, then you cannot discard it and revive it from the graveyard, you will need to resummon it properly first.