What does it mean when she smiles at me a lot?

she does like you, so don't worry about that. but it sounds like she might be going through a hard time with her child since the father isn't in the picture. maybe do some small things for her at work, like bringing her coffee. instead of just smiling and saying hi to her, try talking to her. also, you must understand that you're going to have to move slow with her, because it sounds like she's also afraid of another heartbreak, and possibly another child, which is why she made the excuse of staying home with her child.