Can card effects affect Star Eater between its Synchro Summon in Main Phase 1 and its attack in the Battle Phase in Yu-Gi-Oh?

Answer 1

Yes. Celestial Dragon simply prevents things from being used in response to the special summon, ie, you could not use Torrential Tribute in response. But once this 'window of response' closes, then effects can be activated once again.

The turn player cannot leave the Main Phase unless the opponent agrees to it. That means that if he places Celestial Dragon down on the field and immediately declares an attack with it, the turn has to go back to the main phase if the opponent says he had something he wanted to activate. It can sometimes be tricky for a judge, in cases where players don't declare their intention to change phase, and then a situation arises where one player has declared or revealed something, giving his opponent knowledge of the action, and then the opponent makes the turn go back because he now suddenly wants to activate something based on that 'future' knowledge. But in this case it's easier to handle.

That means that the opponent can technically start as many chains as he wishes, in between Celestial Dragon's summon and response window, and the time when it would be able to declare an attack.