She's normal!
At this age, no, they don't know "how" to react to feeling tired.
A great book is: "what to expect the first year" which you can find at any bookstore or online.
Your baby is real good. AND she is normal. She is sleeping pretty darn good with 4-5 hour stretches. And yes, no baby, just falls asleep as soon as their hits the pillow. NO adult does that either.
Too young for sleep training! No need for that. She is a baby... a normal good baby.
Really. Don't worry.
Both my kids had all sorts of ways and noises and expressions before they actually fell asleep. Normal. Developmental. And per their personality. And no, just because they are rubbing their eyes & yawning... it does not mean they are "struggling"... it is often the way a baby relaxes and goes into falling asleep.
Keep in mind, that maybe she does not want to be interrupted when she is falling asleep... ie: maybe not rub her head and all that stuff. Let her be. My son, would actually get IRKED at me, if I interrupted his sleep and falling asleep mannerisms by my singing to him and rubbing his forehead etc. It sort of irritated him more. Then I discovered... that his pre-sleep noises were just that... and instead of my enabling him to go into HIS own relaxing & falling asleep mode... I was preventing him from doing that. Thus, him not falling asleep well. Because of me. He was able to fall asleep... on his own... and had a pattern of just making noises, grunting etc... then he'd fall asleep. So.. .the moral is: you need to know your baby's cues and what their various noises/cries mean... because, not all cries are the same and not all warrants intervention, from us. Sometimes it is just transitory "tired" pre-sleep cries which a baby does in a rhythmic way... and sort of like zoning themselves out... before they sleep.
--My daughter in comparison, LIKED to be rubbed as she fell asleep. Each child being different.
Keep in mind... that a baby's sleep patterns are NOT static nor set in stone. At each age juncture and developmental change and hitting milestones and teething and change in motor-skills etc., their sleep ability WILL change. They change. Sleep changes. It is NOT static...
Even toddlers, teens, college kids, middle aged people, elderly... ALL change in sleep patterns and ability... and THEY do NOT sleep the same way since they were a baby. If a grown up cannot sleep or has stresses... we don't 'train' them. We help them.
Same for a baby.
Go by your baby's cues... and personality. Not just generic methods.
Really get to know your baby's noises/cries and what it means.
Oh, and 3 months is a growth-spurt time... their intakes needs increases and the frequency of it. Every 3 months, is a growth-spurt time. Feed on demand.
All the best,
Susan