This can be demonstrated by how having items in your cart will not transfer between different browsers (which do not share cookies), and also how if you clear your cookies (specifically one they set called session_id), then your basket will be ...
There is no need to have your website hosted on Amazon.
You can use S3 to store your audio files and then use the public links such as https://s3.amazonaws.com/bucketname/audiofilename inside your website to point to those static files.
Check how to upload files to Amazon S3 (official guide) and how to generate public links.
Not sure what you mean. Are you talking about offline search, or querying a server ?
If you have a server, you're probably good to go with a SOLR server + querying it with jQuery.ajax calls.
If you're looking for an offline solution, you can either do iterative regex lookups or use something along the lines of fullproof http://reyesr.github.com/fullproof/