Can the ArcGIS JavaScript API be installed locally?
Yes. See the Get the API help topic for details.
Can I use a framework like JQuery with the ArcGIS JavaScript API?
While the ArcGIS JavaScript API is built on top of Dojo, you can use a JavaScript library like JQuery in your applications. If using a JavaScript library other than Dojo, you may consider using the ArcGIS API for JavaScript Web Optimizer. The Web Optimizer allows you to create a custom build of the ArcGIS JavaScript API that contains only the modules specifically required by your application. A custom build of the API is small, so it's a great option for those who wish to use a framework other than Dojo. The SDK contains a sample that shows how to work with the ArcGIS JavaScript API and JQuery.
Am I allowed to modify the ArcGIS API for JavaScript?
No. The code for the ArcGIS API for JavaScript cannot be modified.