How can you tell when new contacts come in?
Contact search. Miscellaneous > Date Created Interval.
That depends on what “come in” means for you. If it refers to getting created in the UI then reports are about the limit, however, if from a third party source like a website, integration, Zapier etc… then there is a create/update trigger when you click ‘application’ under Admin → Settings that can be used to raise tags with or do other things like run action sets etc…