I can give you my non-expert opinion, and I'm sure that if I get something wrong, someone will be along shortly to correct me.
Jpeg (Joint Photographic Experts Group) is a file format that uses a grid of pixels to represent an image, so it's a bitmap format. There are several bitmap formats that make them more or less useful in different circumstances. Because the amount of detail in a bitmap image is dependent upon the number of pixels it has, bitmap images can have very large file sizes.
One way to reduce the file size is through compression. Compression can be done by at least two methods: Lossy and Lossless. With lossless compression, contiguous ranges of same-colored pixels that may be described as white, white, white, white, white, black, black, white, black without compression can be described as 5-whites, 2-blacks, white, black with lossless compression. No data is lost, but the reading device needs to be able to read the compressed data and uncompress it.
The other way is called Lossless, in which groups of similarly-colored pixels are changed to a different color in order to create contiguous areas of same-colored pixels. So, if 100 was black and 0 was white (and numbers in between being shades of gray), you might get something like 100, 0, 100, 48, 49, 47, 48, 49, 100, 0, which when compressed might change to 100, 0, 5-47s, 100, 0, and the detail of the different colors is discarded from the file and lost (therefore, "lossy).
With Jpeg, there is also the matter of areas of high compression leaving visual artifacts, which can be seen in the cat image in the Wikipedia page on Jpegs that I linked to above. Many people feel that jpeg is useful for situations where you need smaller file sizes (like low-capacity cameras and slow-connection web viewing), but are not good for printing where file size may be less of a concern than image quality.
EPS (Encapsulated PostScript) is PostScript code (PostScript was originally designed as a protocol for computers to talk to certain types of laser printers) that has been encapsulated with the addition of extra data in order to allow that code to be used as a graphic format. PostScript data can contain bitmap data as well as Vector data (describing shapes with points and connecting lines and curves rather than a grid of pixels). If an eps contains only bitmap data, there is little reason to save it as eps, because modern workflows don't benefit from anything that eps could add, and other formats are usually recommended instead. Even if you have vector and bitmap data in the same image, eps is considered an older format that is better to avoid. PDF Tiff and PSD (Photoshop Document) are formats that can contain vector data and bitmap data.
So, if your team are using eps files of whole ads, how are they doing it? Are they placing the eps into an image frame and using the frame's edges to crop out the unwanted areas? Are they opening the eps in another program and either cropping or removing parts of the ad and then saving? And if the latter, what editing programs are they using and what file format(s) are they saving as? As far as the images shifting, there could be reasons why it's doing that, but I could only speculate without seeing your files and your workflow, which I can't really do over a forum.
I was able to locate a particular piece of said artwork that was utilized in one of our ads this past week. It was a .eps. I, instead, re-saved it as a .jpg, then relinked the file in Indesign,
Where are you resaving the .eps from? Photoshop? Can you open it in Illustrator and save to PDF from there?