When you send any file to print, they should be in CMYK, not RGB.
If you send a file to a client, there's no real way for them to see a 100% scale proof just by a screen representation. I'd in no circumstance release a PDF to the client, on the basis that you are then effectively providing them with your artwork. If they're devious, and through experience, I have come across many of these kinds of people, always send a low quality watermarked version to them. Or, alternatively, alter your pasteboard to A4 and create your business card within. Output a low resolution JPG. Paste this into Word and email the file to them so that they could then print the A4 on a bubblejet (this would have to be in RGB) to get an idea of the size.
As for the printer, always output a CMYK file, as mentioned, with 3mm bleed, and all the fonts outlined if you do not include them in the zip file itself. This will ensure that the printer does not have any missing font issues on their ends.
Also, make sure you only ever create black text with just 100% black, as if the plates during print move, you'll get an eerie blurred effect.