Question Need a help of Room Event

The first two events in the section with the birthdays are events that occur when you
-Unlock every module
-Get at least a Gray clear on every single live quest

Thankfully there is no event connected to getting the Golden Clears ^^;

The third is an event from feeding them taiyaki

 

Yes, from what I know using the DLC key will work for the modules event.

Also yes, you must get at least a grey clear on the challenge live quests as well.
However, you can use norma break modules (check the list on the wiki to see which ones have this skill ^^ ) and that will remove the challenge items.
In most cases, it may make it more difficult to meet voltage requirements than if you used a module with skills to increase the voltage multiplier more quickly, but as you only need a gray clear and not golden that should not be too much of a problem.

No problem~