Well, no, but that's only because we need to process them so that the text is available for search. And OCR is the most reliable way to extract the text from a PDF, which can be structured in any random way.
Not currently
At the moment the only way is by opening the dev tools, under the Help menu
If you open the dev tools and filter by OcrService you will see what it is doing, including how many resources have been processed and how many still need to be done: