I think you'll find the answers to your questions in the Development Lifecycle Guide. There's also a Trailhead module called Application Lifecycle Managment that covers similar material which should be helpful to you as well.
CDN stands for Content Delivery Network. It's basically a caching option to help improve page load times and site performance, as well as your org avoid reaching bandwidth or service request time limits through the use of the
<apex:page cache="true" expires="900"> setting (values are example of page that expires in 15 min). This setting only applies to Force.com Sites since SSL sessions and pages requested after authentication are not cached.
API stands for Application Programming Interface which allows secure programmatic access to your Salesforce org's information; both outbound and inbound to connect with other organizations. The Salesforce1 Platform API Services Guide provides a description of all of the API's and some example integration use cases. You may also find Integration Patterns and Practices useful as well.
The answers to your Sandbox testing questions are greatly dependent on the size of your organization, it's resources and the type of changes you're making. Again, I highly recommend the Trailhead module I've referred you to above along with the Development Lifecycle Guide. I don't believe there's a "One size fit's all answer to your question". I suggest you review the material, then come back and ask questions that are specific to your situation and scenario.