Affinity files are not AI. AI stands for Adobe Illustrator, a very different thing altogether. Affinity files are .afdesign which won't be recognised by anything.
To "convert" an Affinity Designer file to a PDF you "Export" it.
In the menus, go File > Export and select PDF.
Because you are having it printed, in the Preset: dropdown menu select PDF (for print) as shown below, which should be fine.
Click on Export and that will open the filer and create a PDF for you.
As for converting to CMYK that should not make any difference to how well the PDF works, but it might make a difference to how well the colour of your design matches when you get it printed. I would suggest you have a proof made for the first few designs you try.
