Does pulling out baby teeth really make permanent teeth grow faster?

Teeth. Kinda. The permanent teeth will not grow any faster; however, if the root of the tooth is 2/3 of the way grown then it will erupt quicker once the baby tooth is removed. If the baby tooth is removed too early and the permanent tooth has not developed 2/3 of the root then you will wait a long time for the permanent tooth to erupt.
No. Some children lose their baby teeth secondary to trauma, and they do not have any earlier eruption of adult teeth.
Complex. Order and timing of eruption of permanent teeth a complex genetically determined sequence of events. Depending on many factors, extracting baby teeth may enhance or restrict permanent tooth eruption, and may cause malpositions of adjacent or opposing teeth. Best space maintainer is a healthy baby tooth.
Absolutely not. Baby or primary teeth must stay until they can fall naturally on their own. They do not just help to chew food . They also keep the space for the permanent teeth to come in. It is very important to keep them clean and healthy until it is time for them to fall out naturally.
Yes and no. Removing primary (baby) teeth will not make them "grow" faster. However, if the permanent tooth is developing off at a slight angle and the roots of the baby teeth are not resorbing (dissolving) normally, these baby teeth may be considered over-retained and removing them often allow for the permanent tooth to "erupt" faster into the mouth. On the other hand, leaving them alone may delay eruption.