I do think what you describe is very unusual behavior and more self-awareness than a toddler usually has. How does she know to describe her feelings as sad? How does she know the word sad? How did she learn to put herself to bed when she feels this way?
My mother went to bed when she was sad and I learned to do that as a way of handling my emotions. I didn't put myself to bed as a child. My mother may have done so but I don't remember. Have you put her to bed as a way of handling emotions?
I read books about feelings to my granddaughter, starting around that age but I don't think she actually described herself as feeling sad until she was 4 or 5. She would say she was sad if someone asked her if she was sad.
I do know that young children are not always happy and that just as with older children and adults there doesn't need to be anything identifiably wrong for them to feel sad. They also pick up on the feelings of people around them, they can appear to be sad when they're bored, and things to happen to them to cause them to feel sad. For example, they can feel sad after watching a movie or having a story read to them or when thinking about them later. They can feel sad because there is no one to play with them. She certainly does feel sad when she remembers the time when she was the only child. Have you asked her why she feels sad? How does she answer?
Could she be aware of identifying her feeling as being sad because she's heard you or someone else describe either her feelings or their own feelings as being sad? She sounds bright and would pick up on what is modeled by others.
Lying down when feeling sad and getting up 10 minutes later feeling happy is an excellent way to deal with her sad feelings! She is successfully self-soothing.
I suggest that some of her sensitivity and her longing for extra attention is related to now having a baby brother. Each one of us is born with our own personality and her personality includes self awareness and a sensitivity to feelings. This is not unusual. It's just unusual that she's aware of her feelings and found a way to manage the feeling at such a young age when she hasn't been taught to do so by you and/or her Dad. Perhaps she has a sensitive care taker or relative with whom she is often?
If your home life is stable and most of the time peaceful and loving then I wouldn't be concerned about it. It is good to be aware that she is a sensitive child and able to gear the way you parent to her sensitivity. Some kids need firm voices and definite consequences to learn how to behave. Others are sensitive enough that a gentle reminder will put them on the right path.
From your description of how you respond when she tells you she is sad I believe that you're handling it well. Unless she is sad most of the time I wouldn't be concerned. But if she's mostly sad or sad every day, I'd consult with a child psychiatrist.
When you say that you feel "empty inside" when she tells you she is sad, I am concerned about you. I've spent much of my life being overly sensitive to the feelings of people close to me. In therapy I learned that doing this usually indicates that one has poor boundaries. It is good to be empathic and able to relate to the person and their feelings but it's not healthy to "take on" their feelings. It sounds like you may "take on" or feel your daughter's feelings for yourself.
Perhaps you feel sad because she feels sad. You wish that she could be happy and don't know how to help her feel happier. It's important to know that we are not responsible for making anyone else, including our children happy. It is our responsibility to provide a stable home, sufficient food, love and attention, sound parenting practices, that sort of thing so that the child can be happy. However, we are not in control of our child's feelings. Each one of us has our own reactions to life's events. The same situation can cause one child to be happy and another to be sad. Then it's our responsibility to respond appropriately to the feeling while allowing the feeling to remain which allows the child to work out the feeling for herself.
We also need to be aware that the child and the parent are separate individuals and we will not always respond in the best way to how our child feels. That's OK. Both the child and the parent are human and a part of parenting a child is to allow the child to learn how to live in an imperfect world with imperfect people, including themselves.
I learned much of this while in counseling with my foster daughter whom I later adopted. Her birth parent's parental rights had been terminated which naturally caused her to have some very strong emotions. When she was sad, I was sad as I tried to comfort her. She refused my comfort which then made me feel inadequate. When she was angry I worked very hard to teach her how to deal with her anger which eventually led me to be angry because she would only get angrier, What I eventually learned to do was to tell her I loved her and then leave her to work out the feeling for herself. This is what you're describing as you doing when your daughter goes to bed when she's feeling sad.
What complicated the issue with my daughter is that even tho she didn't want my help she also didn't want me to leave her. I had to learn how to stay emotionally uninvolved while still being there for her emotionally. I had to separate my emotions from her emotions. I had to put aside my emotions that were triggered by her emotions and then deal with my emotions later. It can be very tricky.
Eventually I learned better ways of protecting myself so that I didn't have my own strong emotions triggered by her emotions. I'm trying to describe for you the possibility that you may need to learn how to do this. It's very complicated and you may want to talk with a counselor about the way you feel when your daughter feels sad.
For me, my daughter's feelings often triggered my feelings from the past. She was dealing with a sense of abandonment. My mother was often ill and emotionally unavailable to me when I was the age that my daughter was when she was expressing her emotions. Once I realized this, I worked on my own feelings of abandonment with a counselor. When my daughter was 10 and sobbed with a broken heart I also felt that my own heart was broken which made it difficult to respond just to my daughter's feelings. I would get my own experiences and feelings tangled up with hers. I did too much talking. The result was that she felt that I didn't understand and that I was thinking more about myself than I was about her.
Now, I'm able to just sit and listen. Rarely do I even verbally sympathize. My sympathetic listening is all that she needs to resolve the situation.
Your daughter is nearly 3 and probably needs something different than my daughter needed at 10 but this is the idea. Realize that your empty feeling is the result of your own life experiences. Deal with those and you'll feel less empty when your child is sad.
You are doing a good job allowing your daughter express and deal with her feelings. Now, perhaps you need help learning how to deal with your feelings that are brought to the surface by your daughter's feelings.
Also, you probably are not doing anything to cause her to feel sad. It is normal for her to feel sad. It's just not usual for someone that young to be so aware and know how to manage it. You may have even, without knowing it, taught her how to do this. You have, at the very least, provided a healthy environment for her to mature in this way
I just noticed you used the words "emotionally intense" when describing your daughter. Keeping in mind that she does have a baby brother and the usual life of a toddler to stress her, is she more emotionally stressed than is healthy? Life, even for a toddler, is stressful. She is learning new things everyday. Change is stressful, more so for some than others. You can see the ways she's grown and changed quickly in her short years. Good changes, to be sure, but still stressful. Some personalities do have intense feelings while others don't.
I suggest that if it seems to you that she is more often sad than happy and that this combination is noticeably different for her than other children that you should talk with a child psychiatrist. Talk with her pediatrician about this.