1. The Burp API allows extensions to overcome web application hurdles. Extensions can use IMessageEditorTab to decode custom encodings, processHTTPMessage to handle signed requests, ITab to provide new views of an application, and doPassiveScan to automate tasks with new scanner checks.
2. Common uses of extensions include decoding non-standard encodings, automatically signing requests, tracking unique response headers, and detecting encoded values like internal IPs in cookies.
3. Extensions are created with minimal coding by using Burp's extension interfaces to modify requests and responses, display content, and trigger passive scanning checks.