The Community version is free. No you don't have to display the Alfresco logo.
How the page is done is up to you! Rather a broad statement … Alfresco provides a broad set of content management services that you can mix and match to use with an architecture of your choosing. Some sites will have their own front end and use Alfresco for the "content management" part. For example using Alfresco to approve and erving the images behind the "thank you gifts". Equally you if you look at Alfresco share you can see examples of the "news headlines". I you look at Alfresco's web quick start there are examples of managing the layout and content of the web site through some of the content services of Alfresco.
Yes you can create modules called AMPs in the Alfresco world. It rather depends on how much configuration you do as to how you package it up.