There are several issues here.
One is software related, the other is a methodology one.
I will address the software first. It is the easy one, but I'll be mixing tips for the methodology.
1. Source files.
A JPG photo, a raw image from a camera. But also Excel files, word files. Please avoid text on emails or WhatsApp messages as source files.
2. Working files.
PSD, AI, CDR, INDD. But also Excell files, Word, etc. If someone is working on a project, let that person work on the project. If that part of the process is not done, do not send it to someone else yet.
3. Output files.
A pdf is NOT an exchange file. It is NOT about the program reading the PDF, it is for specific characteristics of the PDF files that fixes stuff inside precisely so this content does not move. It is the output format for vector files.
Forget EPS. It is an archaic format replaced by PDF.
For raster images one option is TIF, another is if you really know what are you doing, PNG or JPG.
A flat PSD can also be an output file. Gimp can read them, PhotoPaint can read them.
4. Exchange files.
Here the point is what can we read from another file.
Detailed effects and specific filters can potentially not open even if you are using not only the same application and version if you simply do not have a specific plugin. So separate a base file from a effects file.
You can also do not have a working file if you have fonts missing.
Corel can export and import AI files. Just check the version you can open and save. Try to use the latest one your applications can work with.
CorelDraw Export dialog.
Do NOT use converters! pdf2Cdr!!?? When the application can natively open them!
Gimp can open PSD files. PhotoPaint can open PSD files.
But we can make a long list of files that can simply be opened with different applications. That is your homework.
But you have a bigger issue.
Why a price list should be sent to a "design" department... the design department should make a template so the admin one just works with it.
Why do people not simply use Excel for tables? Open office can use them, MS office, AI, Corel Draw, Indesign.
You need to define workflows, probably by department, by function.
Define priorities. And make decisions.
For big projects probably limit to some applications, probably Adobe. If ppl need to switch just do it.
For middle local projects, limit the applications on that local project, but use professional ones. CorelDraw, Illustrator. Do not use a wide range if you want consistency.
For a one time project, for example, a Facebook post, use the smaller programs like gimp.
Define your workflow!
Some parts of the process are open, then you have a stage where you decide which application, person or department is going to do what. You still have some flexibility on a part of the stage. But there is a point where you stay with the process. and then you generate an output file.
This process can be nested, can be big, can be small, can be repeated. But define them!