A Phantomhive butler does not succumb to a little sleepiness. The last time people were falling asleep all over the place Sebastian's method of dislocating his own thumb to stay awake didn't quite cut it, in the end, and as the days pass it becomes clear he'll have to up his game.
Where would it be absolutely fatal to fall asleep? Deep into the forest. There's little point to focusing much energy on his various jobs anyway, with everyone else on the island asleep-- thus, this week turns into one devoted to Finding A Way Out for Sebastian. It's possible to run into him as he exists the forest, immaculate as always and with no hint of having fought monsters or having fallen asleep on the forest floor at any point.
As soon as the threat of being mauled is removed, however, he does pause to run a gloved hand over his face. So sleepy...
2.
In fairness to Sebastian's stubbornness, it's near the end of the week before anyone has a chance to stumble into any dream of his.
Demons aren't prone to worrying, and they don't truly have any guilt or subconscious issues to process while dreaming. Sebastian's dreamscape is a white, spacious room, flooded with light. It's pristine with a warm and inviting air, but the walls, floors... they're all completely empty. Aside from one firmly closed door in a corner, there's nothing and no one there besides Sebastian - who was busy using a feather duster on a nearby wall - and he turns towards the visiting person with a smile.
"Welcome. How do you prefer your decorating?"
He'll adapt the dreamscape according to your preferences; no problem.