There is no DOM access of any kind from a web worker - period. ALL DOM manipulation must be done from the main JS thread. Web Workers can only communicate with the main thread via messaging.
This previous SO question tells you that web workers cannot access local storage, something that was easily found with a Google search.
That same thread has a list of what web workers can access.
It is unclear what you mean by "include a file in a worker". You can import scripts. I don't know what else you're asking about. You can use an ajax call to fetch data if that will help you with your audio file.
This sounds to me like you need to do some basic research about what you can and can't do with web workers (as there is a TON written about it on the web) and then come back with a lot more specific questions that incorporate that knowledge.
There are no hacks for web workers that let you do more than you are supposed to be able to do.
You can do work in the main JS thread and do it in little chunks with setTimeout if you have a lot of work to do and want to keep the main UI as responsive as possible. This is a very old design pattern that goes back to the days before threading was available or easy to use. It involves designing your work such that it can be done in little pieces with state stored in some persistent object and then you can repeatedly do small amounts of work and then return back, only to pick up the next piece of work on the next timer tick.
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