Without seeing a plan and knowing sizes I will hazard to say that the wall space between the two windows in the last photo is seemingly the most logical spot for the bed. This placement might now allow for nightstands as one leaf of the closet door would probably bang into a nightstand on that side. It seems to be a very large space. You might be able to accommodate a seating group between the bed and the office space alcove. The alcove where the bed is shown in the photo might lend itself to a window seat and perhaps narrow cases or shelving for books.

IMO small rugs beside the bed work best in this case since without pics I have no idea how this space is even layed out .Not exactly a huge space for 2 queen beds. I think there would be very little of a rug even showing in the space so post some pics of the space with the beds in place.

Larger or smaller. With a daybed, usually you see either: 1. rug floating in front, the same width or a bit narrower than the bed or 2. rug underneath, and 24" or more wider on the sides and several feet in front. Measure out with painter's tape. The large rug option will not slide around. You need to add a felt rug pad designed for rug over wood in the proper size. Or you can get a rug that fills the room to within 8-24" of the walls, also with a proper rug pad. If you go with a smaller floating rug, choose a round or irregular shape.

What a lovely baby girl's room! Your attention to detail and thoughtfulness in its design are truly impressive and the artworks you made on the walls brings a sense of joy and playfulness to the room.