I have to comment on the fact she is still in a baby bed. The bed is more than likely uncomfortable now. They are not made to support kids this big. She needs to be in a twin, full, or queen size bed with real sheets and blankets tucked in at the bottom. She is way too old and big to be in a crib.
And don't even consider a toddler bed. They are built for kids that are toddlers, she is the age of a pre-schooler. Toddlers are from ages 12 months up to 24 months.
That's all, just that is surprising to me.
As for giving up the nap, she's too young. If you are considering sending her to pre-school in the fall she will be required to nap during the day. They will not let her stay up, she will have to lay on her cot. Even in pre-K they have to take naps and if they don't they just lay there and rest. It is required by the states regulations. Kids under 5 years old are required to have rest time at school. Most schools have requisite nap time after lunch.
I would just let her learn how to put her own blankets over herself. She is just wanting a bit of attention so she does this. I think if she wants it and you are okay with doing it then keep doing it. It gives her comfort. But I would move her baby bad to the side of the room and put at least a full size bed in her room. She will find it wonderful.
I would put her in the big bed then get past her getting used to it, then start working on the not waking up.