baby is always tired

I have a daughter who is now 2 1/2 years old. In the past 2 months she has started to complain about "being tired". It mostly happens when she is bored ( i.e. when I do housework, cook or sew, all activities that draw my attention away from her ) She wants to lie in bed and wants a pacifier. If I do not take her to bed, she rests on the floor or on a pillow. She just stays there 10-15 minutes, usually talking to herself or playing a bit . she seldom falls asleep under these circumstances.
I am worried about the possible causes of this behavior.
When we get out she wants to be carried a lot and always complains she is "tired" - but this seems quite common among toddlers. Also, when I wake her in the morning she often refuses to get up. She says that she is tired and that she wants to stay in bed a little more. A possible cause of that may be that I am trying to wake her earlier b/c if I let her sleep too much in the morning she will mess up all her schedule and end up going to bed at 11.00 PM , thus waking even later the morning after, and so on.
I am confused, I think she might be somewhat depressed? She does not see other children very often,and I do not have the time to play with her as much as she would like to. Also, lately she has been a lot more irritable, with tantrums and nonsense behavior, amd aggressive bahavior towards some of her toys, too.
She is becoming more and more dependant on my presence and I cannot leave even for 1 or 2 hours to do greceries because she goes crazy. Instead of weaning her off pacifiers she is sticking to it more than ever.
Do you have any clue? could that be some chronic illness? Or rather a psychological problem? I am worried and have no clue. Thanks.