First off, if you are working at a printing company you should already have some guidelines on what profiles you should be using. Standardization is everything is this industry.
That being said here's my breakdown for your questions.
1. SWOP vs FOGRA
They are different standards that have been created to work with the most printing presses. They are developed by testing a large number of presses using a range of paper types and produced a set of datasets.
The main difference is that SWOP is used for printing in the US and FOGRA is used for UK/EU printers. That being said this is just how each profile is GENERALLY used, there can certainty be printers in the EU using the SWOP standard.
2. The best color profile I should use when working on CMYK files
If you are being supplied images or artwork the client SHOULD always specify what color space they need. If they don't that's a whole other talk about project management.
Most will suggest ISO Coated v2 or Coated FOGRA39. Others will have different international standards that they conform to and have profiles they can send you. A few will still just say CMYK and not know enough about color management to be more specific. If that is the case then ISO Coated v2/Coated FOGRA39 would still be the safest choice.
3. Should I use the same color profile for Illustrator as Photoshop
YES YES YES YES. You should ALWAYS ALWAYS ALWAYS work in the same color profile from software to software. I can not stress this enough.
Bonus:
I'm not really sure why you are going through this trouble for a QR code, your rip software (software used to send files to the printing press) should be set to the correct profile for printing.
As long as you aren't trying to send RGB art to the press the software should strip any profiles associated with the art and use the correct ones for the press.
Additionally, what file format is the QR code? If it's anything remotely vector editable you can just manually select the art and change its color. If it isn't you can vectorize it very easily.
