Will the red, blotchy rash on my newborn’s face go away on its own?

Yes. Newborns can have funky rashes like the one mentioned. They can be a result of hormone withdrawal and do go away on their own. Neonatal acne can also be present. Just clean baby's face with a warm wet washcloth. Any rash that persists, worsens, or worries you should be seen.
Yes. Many babies have rashes in the first few weeks after birth. One common rash is called erythema toxicum neonatorum. It's a scary sounding name for a completely harmless rash that looks like "flea bites". The spots come and go, are usually are present in the first few weeks of life and then resolve on their own. Of course, if your infant doesn't appear well, you should contact your doctor.