If I convert a pdf file to an Illustrator eps file, will it be a full vector file that will resize appropriately?
The EPS format is merely a container. What is in that container can vary.
An EPS can include 100% raster data. An EPS can include 100% vector data. An EPS can include any combination of raster and vector. The simple fact that a file is saved to the EPS format does not mean it's automatically vector.
Unlike Photoshop, it is possible to create 100% vector eps files with Illustrator. However, much depends upon how any file is constructed within Illustrator. If you use the effects for shadows, or blurs, or glows, you are embedding raster data in the EPS.
So the reality is... An EPS from Illustrator can be totally vector, but it doesn't have to be.
Depends on what the file contains. Unlike a straight raster format like say a PNG file, vector formats can embed much wider amount of data. Raster files included. So no you can not draw the conclusion that a encapsulated PostScript (EPS) is automatically scalable.
If you convert a PDF to postscript via Illustrator you go trough the filters of Illustrator. Since Illustrator does not support all features of PDF files something may go wrong. Also fonts may be problematic
If on the otherhand you directly convert the PDF to EPS you get wider support assuming the file does not contain features not supported by EPS. You can allways use the print feature of your eps viewer to make PostScript that can then be encapsulated.
So no you can not draw this conclusion. But somethong like that may happen. So one could say that it is possible that it does. If its a simple case most likely yes.
As Scott noted, if you use some of the items in the Effects menu for shadows, blurs or glows you are embedding raster data in the EPS file. But additionally, you are adding transparency, which the EPS format does not support, sometimes creating stitch lines and slices during the flattening that must take place. Saving an Illustrator file as an EPS in which transparency Effects have been used may render the file even more unusable and unscalable than originally intended.