You're pretty unsure of these concepts so I'll warn you that you have a lot of learning ahead of you. But, be patient, and take the time to research, educate yourself and become comfortable with all this and you will be much better off.
Now, to attempt to answer your specific questions:
- Using Google App Engine does not preclude using iOS. As long as your web services are generating JSON or XML, you should be fine.
- Hosting is free with a number of these services (others include Amazon's AWS and Microsoft's Azure) as long as the consumed resources is low.
- Databases can be set up in a variety of ways, from something as simple (but not very flexible) flat files to Oracle or SQL Server installations.
- A web service is a communication layer between two devices over a network. Think along the lines of sending a URL to a remote server and getting a response, say in the form of JSON, in return.