I know that brain size is less important than structure and that human babies' brains are 27% the size of an adult's, but humans give birth to large brained babies. As a result, humans can't support the weight of their own heads for a few months, mature slowly and have a large pelvis. If a species had shorter pregnancies (about six months long), larger skulls and smaller babies than humans (almost half their size), could their brain size increase rapidly until adulthood to the point they end up being smarter? In this case, an average adult is slightly taller than a human and their brain structures are somewhat similar, but with better long term memories.
Here is a very little baby. It was born at 6 months gestation. Issues which make such early babies less likely to survive involve lung and digestive tract maturity but there is nothing hardwired into the mammal body plan that means you need 6+ months to get working organs - puppies and kittens this size breathe fine after an even shorter gestation. Preemies this small might grow up developmentally disabled. Or they might be normal!
Your human-types can have very tiny babies like this one, but which have matured their lung, mouth, digestive tract etc in the manner of other mammals to help them survive once born. The babies, their heads, brains and whatever other parts interest you continue to grow ex utero. They get as big as your story needs.
Yes. This is what marsupials do all the time. Marsupials are born at a fraction of the adult size, and are born with very loose skull bones (akin to a fetal stage of a placental) that would allow for the brain to expand much more than a placental mammal that has to fit through a birth canal. Most of the nutrients a marsupial baby consumes come not from the placenta, but lactation. The brain goes through most of its development and growth outside of the womb. Marsupial pregnancies are often incredibly short, going from fertilization to birth in just 11 days.
Marsupial brains are often characterized as smaller than placentals, but the strange thing is that marsupials have a greater potential for developing large brain size than placentals. Exactly why they never took advantage of this is still unknown.
Also, the birth canal hypothesis may not be right. One line of thought now is that human babies are born at a relatively helpless, premature stage not because if they stayed in the womb any longer they would be too large to give birth to safely, but because if they stayed in the womb any longer their demand for nutrients would be a serious threat to the mother's health. In placentals the baby is essentially a parasite on the mother for the earliest stages of its life, and human infants with their large brain demand a lot of calories. Giving birth to the baby gives the mother more control over how much nutrients she gives to the baby and allows the baby to be fed external sources of food rather than just parasitizing off of the mother.