I use two programs for viewing and handling PDFs. One is a simple viewer and annotator (Foxit), and here you can search for a word or a phrase in a single PDF or all PDFs in a Folder. The other is called Mendeley Desktop, which is more of an "iTunes for PDFs" (made for academia).
But if a PDF is an image, obviously it won't show up in the results. If I have a list, I could OCR and convert them.
So, no, I don't have to open each file one at a time to perform a search. Both programs do an indexing of the texts of the collections.