Yes, „w” requires locative here, so „moim łóżku”.

No, that's for feminine. moja (Nom.) --> mojej (Loc.)

Oversight, added.

But in 1st person, the 'normal' possessive also sounds natural. "Śpię w moim łóżku" sounds perfectly fine to me, but then "Ty śpisz w twoim łóżku" isn't that great, it really should be "swoim".