Note that this appears to be Estelle's room (basically you tunnel into it from below), which is why the map is locked out after she is captured.
The G have ridiculously high dodge rate and defense, but very low attack power. The best way to dispose of them is to herd them towards the yellow beacon traps.
G tend to avoid rooms which your characters are in, and move away from your characters. Use this to get them to move into the room with the traps.
If you corner Gs, they may end up not moving at all, so give them some space unless you are trying to kill them in combat.
But if they get the chance, they will make a beeline for your base. Given that they have very high movement, double move and the ability to multiply, things can get dangerous quickly if they're even two rooms away from your summoning pillar. You still don't lose and suffer a game over though, just get kicked out of the map.
Killing any G in combat will net you Special Item 065: Proof of Killing G (which is just a trophy).
The easiest way to kill a normal G is to cast the S Class magic Dark Lance on it then hit it with Lily's Gorogoro attack when she is wielding a high accuracy weapon such as an Elfin Bow.
To kill the King G, you will need to use the Pierce skill, or weapons with piercing or armor ignoring attacks such as several of the spears or the A Class Magic Bow Remiel..
King G takes half damage from all sources due to Godly armor.
King G always has at least 50 PDef/MDef. As max stats is 100, this means you cannot deal more than 1 damage to him without Pierce.
As a BOSS, he cannot be Dark Lanced.
The red beacons are items which can be picked up and put back into place in the furniture around the map
Clearing this stage unlocks EVENT-衣装部屋／おもちゃ箱 which lets you change Lily's dungeon map costume to an alternate one. There is one alternate costume for each growth stage.