I'm running an application that is a web app and the backend for an ios app as well as a database in an ec2 instance. Will these processes resume if I stop, resize, and restart my instance? Will I need to go in and run startup scripts?
It's just like turning your laptop off and then back on. If you have configured the services to start automatically on boot then they will restart, if you haven't configured them to start automatically on boot then they won't.
This is a function of the software you have installed and configured on the EC2 instances, and whatever Operating System your EC2 instance is running. You won't find much about this in the AWS documentation because it isn't really Amazon specific.