Can you please explain more? Resting the head and closing the eyes is normal, but it's hard to visualize without any pictures or at least a detailed explanation of what you are witnessing. Pictures would be better.
Depends on how old the baby is. After they fledge around 4 weeks is when they normally would start to venture out of the nestbox and begin perching. Babies should be kept in a brooder environment with specific temperature and humidity until they are fully feathered (usually around 4-6 weeks) and then put into a training cage.
These cages are usually smaller, have a couple low perches and toys to entice them to climb around. Then when they start weaning (8-12 weeks old) then they get better at perching and go into a normal sized cage with lots of perches and toys. They learn to sleep on the perches.
Baby birds follow same principles as any other living thing sleeping. They will close their eyes. Until they learn to perch they will sleep on the ground in all kinds of ways, sometimes they will lean against something soft or just sit up and turn their heads around to tuck their beak into their shoulder. Sleep is good and normal for babies.