Hmmm, the built-in bookshelves in the second bedroom does lend itself to being the office, doesn't it? And since you are both in school currently, it is very important that you two have a comfortable space to study in.
I still would like to see a larger playspace for your daughter, though. The truth is, you don't have enough space for all of the functions that you want, so something will have to give. Have you looked at IKEA stuff yet? I'm not talking about buying anything there, but just that they have a lot of great ideas for multipurposing small spaces. The living room can be set up as more of a playroom, without the need to have just one corner be a play area, if you incorporate a theme of fun and playfulness throughout the whole living room. In other words, a playroom for DD that also happens to have adult seating.
If you need more storage space, consider lining the dining room walls with shelving (square shelving in very dark brown would be very chic). This could be used to store toys, art/craft supplies for DD, and/or dishes, and display more of those photos, and all of it can be done very attractively. I love your dining table set, btw.
Our living room has a train table and a large basket for toys. I also keep a basket of wooden blocks under the futon, along with a potty chair. In the dinning room, DS has his own small book shelf which holds books, craft/coloring supplies, play dough, musical instruments, and play food. Also nearby is the play kitchen that I made him for Christmas last year. Our office has a basket of kids books for DS, as well as a toy barn with animals. In our bedroom, DS has another basket with books for bedtime reading. Then there is also another play area in his bedroom, but since it's on the second floor and he's rarely in there, he doesn't play in his room very often. I rotate toys to different rooms every so often, so that he doesn't get bored. If you want to see pics of these play area, let me know and I'll post them.
I think making play spaces around the house is fine and logical with a small house. A designated play room is a luxury, IMO. Plus, not all children had the temperament to play in a different room by themselves.